I know this is as tough process, but you need to know that it's also a LONG process. If she has Gout, which I strongly suspect and am almost certain that she does, she isn't going to show much improvement until you get her started on the Allopurinol daily to lower her Uric Acid levels. Gout is extremely painful, that's why she is acting so lethargic and cannot move around very well. She's in a lot of pain from the swelling, and then the extra strain on her kidneys is also causing her to feel sickly. It's mainly the pain in their extremities/joints that causes them to feel so badly and be so lethargic though.
We just helped a woman on here who pretty much followed the same path you are, she took her dragon to a vet, they didn't do any blood work and just put the dragon on antibiotics without running any diagnostic tests at all, and the antibiotics not only didn't help her (because she didn't have an infection), but it made her worse because antibiotics are very harsh on their kidneys, which were already very stressed by the Gout. So we talked to her, she took the dragon back to the vet and had to demand that this vet (exotics vet, not a reptile specialist) take blood and run a routine blood work panel, because the vet didn't have any idea that Gout was a common problem in Bearded Dragons and absolutely denied any possibility that the dragon had gout. But she stood her ground and demanded that the vet run blood work, and then they had to wait another few days to get the results back, and her Uric Acid level came back at 40!!! (10 is as high as it should be!!!) Then she came back on here and told us that obviously her dragon had Gout, and we just assumed that the vet had given her a prescription for Allopurinol and pain medication...nope, this vet decided that she would have to ORDER Allopurinol from a Veterinary Medicine Supply Company that came shipped already compounded in the correct dosage for the dragon and already in liquid form, and it was going to take another 2 weeks to get the Allopurinol!!! And, this poor beardie owner had already paid this vet $230 for a 2 month supply of the Allopurinol from this company, because she wasn't aware that any Reptile Specialist Vet would have simply written a normal prescription for the correct dosage of Allopurinol compounded in liquid form, and she could have taken it to any pharmacy and had any Pharmacist compound the Allopurinol in the correct dosage and into liquid form, and that a 2 month supply this way would only cost an average of between $15-$25 maximum!!! So we had to fill her in, she called this vet back fuming mad, and this vet acted like she had no idea that she could just write a prescription for Allopurinol, and in my opinion she lied to her, telling her something like "You have to give her "Veterinary-Grade" Allopurinol, not Allopurinol compounded from Human-Grade tablets from a human pharmacy, which is a bunch of BS because THERE IS NO VETERINARY-GRADE ALLOPURINOL!!! They don't produce a Veterinary-Grade Allopurinol at all, the stuff that this vet ordered would have had to have been compounded from the same Human-Grade tablets that are kept in any regular old pharmacy. So either this Veterinary Medicine Supply Company or that "exotics" Vet was trying to make a bundle of money, as again, that 2 month supply of Allopurinol she had ordered and charged this poor woman $230 for would have only cost a retail of between $15-$25, and probably had an actual wholesale cost of around $5-$10! Quite a profit margin there...so we told her to call the vet back, demand that she cancel that order, refund her $230 back onto her card, and write a prescription for the Allopurinol so she could go pick it up immediately. I think it still took a few days longer than it should for this vet to agree to do this, but she finally got the Allopurinol prescription called into her regular pharmacy, and her pharmacist compounded the correct dosage for her dragon and put it into liquid form, and she paid like $20 for the 2-month supply.
From that point it still usually takes a couple of weeks of them being on the Allopurinol for their Uric Acid level to drop enough to show marked improvement, but finally after I think about 3 weeks or so of being on it her dragon perked back up, started eating again, moving, wasn't as lethargic, etc. The Allopurinol is amazing stuff and works so very, very well, but the key is getting them on it as quickly as possible. This is exactly why I told you to "Stand Your Ground" with this vet when you go to the appointment!!! You must absolutely demand that they take blood on that day during that appointment and that the blood panel they run include a Uric Acid level, a kidney panel, and a liver panel, along with the routine CBC. This will confirm or rule-out not only the Gout, but also any type of infection, and will show what her kidney function is like and how much damage, if any, has been caused by the Uric Acid (it's reversible once the UA level is dropped).
The bottom line is that the longer a Veterinarian is allowed to dick-around and not simply perform routine diagnostic tests, the longer it will take to get a proper diagnosis, the longer it will take to get the dragon on the correct treatment, and most importantly THE MORE DAMAGE THE VET CAN DO BY JUST SKIPPING DIAGNOSTIC TESTS AND INSTEAD "GUESSING" WHAT MIGHT BE WRONG AND THEN PRESCRIBING ORAL ANTIBIOTICS OR ANTIBIOTIC INJECTIONS THAT WILL NOT ONLY MAKE HER KIDNEYS MORE AND MORE STRESSED IF SHE DOES HAVE GOUT, BUT COULD ACTUALLY KILL HER BY PUTTING HER INTO RENAL FAILURE IF SHE DOES HAVE GOUT....Pisses me off, these vets that make a good income to just walk into an exam room, run no tests, ask no questions about the dragon's husbandry or possible causes of the illness, just look at the dragon and then tell the owners that "it's probably some type of infection or parasite, here, give her this broad-spectrum antibiotic for 2 weeks and if she's not better at that point then maybe we'll do a culture or blood work"...
By the way, did you get the T5 UVB tube/fixture yet? That should perk her up a bit, but don't expect a lot until she gets the blood work done and gets on the Allopurinol and some good pain medication. That's the other thing, be sure to ask for pain medication, like Tramadol. That other member's vet refused to give the dragon any pain meds, I guess because she refused to admit she might have Gout...you must demand the blood work, demand the pain meds, and actually I would request a prescription for the Allopurinol on the day of the appointment. It will take a few days to get the blood work results, but if you already have the Allopurinol prescription filled and on-hand, all they have to do is call you up when they get the blood work result back and say "Yep, she has Gout, start giving her the Allopurinol right now". That's much better than them calling you on a Friday, telling you she needs Allopurinol but they you can't get the prescription until at the earliest Monday, etc. You'll instead have it ready to go, because it's pretty obvious that Gout is the issue....AND GET COPIES OF EVERYTHING, ALL TEST RESULTS, VITAL SIGNS, NOTES THAT THE VET MAKES, DOCUMENTATION OF ALL MEDICATIONS AND THE DOSAGES GIVEN, AND IF THEY DO AN X-RAY OR ANY OTHER IMAGING TESTS BE SURE TO GET THEM TO EMAIL YOU A DIGITAL COPY....