I found and examined three CMSRs for W. F. Richardson. The first was for his service in the 1st Texas. It was a bit confusing due to another individual's cards begin mixed in with his. 1st Sgt. W. D. Richardson evidently was in the same company (L). (There are pay vouchers for both men for overlapping time periods with signatures that definitely show these to be two separate men.) It appears that when W. F. Richardson enlisted, August 1, 1861, he was 3rd Sgt. of the company; however at least by Jan. 1, 1862, he was detailed to "sick camp". Reading between the lines, since he could not take care of the sick and serve as a sergeant at the same time, someone else was promoted to sergeant to take his place, and he was demoted to private. Seems a rotten deal, but at least he got 25 cents per day additional pay. That made his monthly pay total $17.50 as opposed to $17.00 for a sergeant. A detailed letter from Captain Bedelle, Commanding Company, dated September 13, 1862, discharges Private W. F. Richardson from the 1st Texas Regiment "by reason of promotion to the rank of Asst. Surgeon by Special Order No. 215 from the Adjutant and Inspectors Gen'l office at Richmond." Incidentally, an assistant surgeon's pay was $110 per month. I'm sure Dr. Richardson did not object to that!
The second record is for Asst. Surgeon W. F. Richardson, PACS. In that record reference is made to his commission date of September 13, 1862, confirming that this is the same individual as above. However, there is very little in the record covering 1863. Some from 1862, then a good bit from 1864 and 1865, but nothing that would confirm that he served with the 2nd Florida. In the CMSR for Asst. Surgeon W. F. Richardson with the 2nd Florida, there are a couple of cards that do not state 2nd Florida that probably belong in the PACS file. Only one Confederate card in the 2nd Florida file shows him in that unit. It is dated Dec. 23, 1863, from the Medical Directors Office, Richmond, Virginia. All other cards that specifically state 2nd Florida are Union POW cards.
So, unless I missed something, there is nothing in these CMSRs that positively states that the 2nd Florida man and the 1st Texas man are one and the same, but, at the same time, there is nothing that conclusively says they are not. My gut feeling is that they are indeed the same man.