Bryan Howerton has collected much of the sparse information available on this short lived regiment.

Posted By: Bryan Howerton Date: Wednesday, 26 January 2005, at 3:48 p.m.
In Response To: Re: Williams' Arkansas Infantry (Gordon Hale)
Smith's Company was Co. C of Adams' Arkansas Regiment. The company was recruited in Carroll County. The regiment was first commanded by Col. Samuel W. Peel, then by Col. -- Fitzwilliams, finally by Col. Charles W. Adams.
Here is a brief history of the regiment, from an 1897 letter written by Samuel West Peel:
The Regiment that I commanded as Col. was raised in the latter part of the summer of 1862. This Regiment was a Confederate Regiment. I was Col. I think that a gentleman by the name of Hill was Lt. Col. Not certain about that. Mr. James H. Williams of Carroll Co., Ark., was Major. Soon after its organization I was at my own request reduced to Lt. Col. and a man by the name of Fitswilliams was appointed Col. and Mr. Williams remained Major. After Genl Hindman took charge of the T.M. Dept. he reorganized the Regiment by the appointment of Charles W. Adams of Helena, Col. Fitswilliams was reduced to Lt. Col., Major Williams remaining Major. This Regiment was called the 3rd Ark. Infantry. At the Battle of Prairie Grove, Ark., the Regiment became scattered and was attached to other commands. The Regiment when commanded by me was composed of ten companies. The Captains were well known to me then. Will give all I can remember. Capt. Hugh Berry, Capt. H. M. Moore, Capt. Larkin Bunch, Capt. Parks, Capt. Combs, Capt. H. C. Dickey, Capt. Williamson & Capt. Smith and J. W. Bishop.
Colonel Peel is being somewhat disingenuous when he states that the regiment "became scattered and was attached to other commands" at Prairie Grove. In fact, Major-General Hindman determined that the regiment "dishonored itself" during the battle, "after delivering a single fire, the greater part of the men broke ranks, threw down their arms, and shamefully fled, many of them even deserting to the enemy." The regiment was disbanded on December 16, 1862, and its remaining members were transferred to other commands, notably Shaler's 27th Arkansas Regiment and Harrell's 17th Arkansas Battalion.
The regiment was mustered into service as the Third Regiment, Second Brigade, Northwest Division, Trans-Mississippi Army. The State Military Board had authorized it as the 36th Arkansas Infantry, but this designation was instead given to Glenn's 28th Arkansas Infantry after Prairie Grove.

In Response To: Fitzwilliams’ Partisan Rangers (Bob Meeks)

From Embattled Arkansas: The Prairie Grove Campaign of 1862 by Michael Banasik:
page 415 Charles Adams ordered his regiment of Arkansas conscripts to charge. They did "in a most gallant style" but were met by murderous fire from Clark's Indiana boys, causing the "greatest consternation and confusion" in Adams' regiment. Disaster followed. Lt. Col. JAMES FITZWILLIAMS bolted from his horse and sought refuge behind a nearby tree, then ran headlong from the front. The troops, following his lead, panicked and thundered away from the field. Col. Adams and Major Williams subsequently rallied about 100 of the regiments 461 men with the help of Corporal Howater, the only uninjured of the regiments color guard.
Of the hundred who stayed, George Maddox never fogot the moment when he stood alone with only one other man of his company - and he was wounded. With no one to provide guidance, Maddox helped carry private Hamilton McChristian from the field to a makeshift hospital at the Buchanon house near Prairie Grove Church. There Maddox met Lt. Col. FITZ WILLIAMS, who told of how he was wounded after having his horse shot from under him three times. Maddox beleived the story and would not learn for several weeks that Williams had lied.
sources for these 2 paragraphs: C.W. Adams report in the OR; Maddox, Hard trials and Tribulations of an Old Confederate Soldier.
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Numerical designation of Adams Ark. Inf.?
Bryan Howerton , 9 June 2007, at 8:23 p.m. In Response To: Numerical designation of Adams Ark. Inf.? (Shawn Clark)

Charles W. Adams was elected colonel of the 23rd Arkansas Infantry on April 25, 1862, and resigned when his regiment was field-consolidated with the 12th and 14th Arkansas Regiments at Saltillo, Mississippi, on September 10, 1862. The 23rd Arkansas was east of the Mississippi River when the battle of Prairie Grove took place on December 7, 1862.
The regiment known as Adams' Arkansas Regiment, the one which fought at Prairie Grove and was later disbanded, was one of three regiments which were initially known as the 1st, 2nd and 3rd Regiments, Northwest Division (Adams' being the 3rd Regiment).
These regiments were authorized by the State Military Board as the 34th, 35th and 36th Arkansas Infantry (Adams' being the 36th Arkansas), but these designations were not used before Prairie Grove. The 35th Arkansas was known then as Rector's 1st Arkansas, the 34th Arkansas as Brooks' 2nd Arkansas, and the 36th Arkansas as Adams' 3rd Arkansas.
The 34th and 35th Arkansas finally took up these designations after Prairie Grove (though the 35th was later resignated as the 22nd Arkansas). Adams' regiment was disbanded after Prairie Grove and its designation of 36th Arkansas was given to the 28th Arkansas. So, while the records of the State Military Board refer to Adams' regiment as the (first) 36th Arkansas, this designation was never officially used.
I hope you can make some sense out of this jumble. This was not an easy puzzle to work out, let me tell you -- and it's just about as hard to explain as it was to research!
One of the legacies of this confusion is that some people still insist today that Adams' 3rd Arkansas (Northwest Division) was really Manning's 3rd Arkansas (Army of Northern Virginia), which commuted back to Arkansas for the Prairie Grove fight. In fact, when the battle of Prairie Grove took place, Manning's "real" 3rd Arkansas was digging in on the heights south of Fredericksburg, Virginia, getting ready to welcome the Union Army of the Potomac.

Governor Rector issued an address on May 5, 1862 calling for the formation of 30 new infantry companies and 20 new cavalry companies. Most of the states' militia regiments had conducted their final recorded militia muster during the last week of February and the first week of March 1862. Rector indicated that if there were insufficient volunteers to fill these new companies, a draft would be made upon the militia regiments and brigades. As a further enticement, Rector also indicated that these regiments were for home defense and that they would not be transferred to Confederate Service without their consent. (The Rebellion record: a diary of American events, with documents ...,

Volume 5 edited by Frank Moore, Page 11. Accessed May 3, 2011, http://books.google.com/books?pg=PA11&dq=Arkansas+State+Military+board&ei=rg7ATb36IIb20gHvu9WOBQ&ct=result&id=1bB2AAAAMAAJ#v=onepage&q=Arkansas%20State%20Military%20board&f=false)

During the spring and summer following this final muster, many former militiamen joined one of the newly formed Volunteer Regiments. It may be that the militiamen decided it was better to enlist and remain together than to wait for forced conscription under new Confederate Conscription laws, which were being strictly enforced during the summer of 1862. Rector's much reduced former regiment, the remaining portions of Companies A, B, C, and G, were reinforced with troops, many of whom were came from the from the 58th Regiment Arkansas Militia regiment of Franklin County, the 15th Regiment Arkansas Militia of Pope County, and the 10th Militia Regiment of Johnson County.
Colonel Rector's new reinforced regiment, in accordance with Governor Rector's plan of maintaining the organization as a regiment of state troops, was initially organized at Fort Smith as the "1st Regiment, Northwest Division, Trans-Mississippi Department" with 1037 men. Also called Rector's War Regiment, 1st Arkansas Volunteers
Two other new regiments were raised under this plan, Brooks' 2nd Arkansas and Adams' 3rd Arkansas. These Regiments participated in the Battle of Prairie Grove under these state designations. When finally inducted into State Service, these regiments would become Rector's 34th, and Brook's 35th Arkansas Infantry. Adams 3rd Arkansas, after breaking under fire at the Battle of Prairie Grove and being ordered broken up, never assumed its intended Confederate designation of 36th Arkansas Infantry and that designation eventually went to McRae's old 28th Arkansas Infantry in the reorganization of the army following Prairie Grove.