I have tendinitis and also tendinosis and carpal tunnel and osteoarthritis. I have tried many of the braces on the market, and the ones that help me the most, especially with the inflammation associated with the arthritis and tendinitis/tendinosis, is what are called thumb splints. They are neoprene braces that slip on. My former Occupational Therapist is the one who recommended them, and she cut the thumb portions down for me, so they would interfere less with my daily functions. I lost one and purchased a replacement online (possibly on eBay, although I don't recall exactly where.) If you're interested, you could do a search on Rolyan and Sammons Preston, they are the makers of the splints I use.
As far as medication, anti-inflammatories and pain medications do help, and I also use a paraffin spa, which decreases the swelling and the warmth soothes the achiness. I also use ice. I believe in a multi-modal approach to healing, so don't rely on just one method to manage my pain and inflammation.