No one is too young for Arthritis, apparently your doctor is not familiar with Juevenille RA or Osteoarthritis at a young age.
Mine started at age 10, or shall I say discovered at age 10. It strikes at any age, especially RA.
It is possible you have a more rare form of arthrtis, fibromyalgia but you didn't indicate that you had any other symptoms. There are also various symdromes that also cause joint pain such as Ehlers Danlos Syndrome which is a genetic Connective Tissue Disorder. Type 3 causes the most extreme pain in nearly every joint. It is characterized by "loose" or hypermobile joints, soft skin, attrophic scars (they look flat and wide, reminds me of the texture of a worm, slow healing and many other key indicators. Not all folks have the same set of markers like soft elastic skin, it does vary and there are many types and two which are fatal. Those types however cause a huge set of problems you'd know of such as heart problems, gastro bleeds and the like.
Some forms of arthritis never increase the SED rate much. Mine is always at or below 10. 100 is a sign of severe imflamation. I can walk into the doctors completely swollen, joints hot and all tests come back negative.
The truth is you could have arthritis, and it does in fact sound like it. There have been cases of individuals who have had false positive and false negative. You may text negative one time then positive the next so it is really important to test often. Especially if on Celebrex you need a full kindey panel every 6 months to make sure your kidneys are properly functioning. When they do that they should do another full work up checking for RA, SED rates and the like.
If you are not happy with your doctors answer you do always have the right to get a second opion. If your doctor is adament that you are too young then yes I'd definately seek a new one. His statement however could have been a generalized one. I often hear "you are too young to be in so much pain and have so many health problems". They aren't saying it's not possible they are just indicating how sat that it is for me.
Not sure what else to tell you, you are on the right medications for general arthritis but if you do in fact have RA or a different for there are other medications you should be taking as well. It is a matter of getting a proper diagnosis first then determining what your next move will be.
Best of luck