The ice and heat won't do much if it's carpal tunnel. It's fluid retention that's causing your discomfort not inflammation. I have it in both hands too. Try to cut back on your salt intake and wear the splints. Sometimes acupressure works for me but it's only temporary relief.
Definitly, i totally understand:( i have arthritis and tendonitis in both my hands, from being a cake decorator and a dog groomer, splints only do so much. I like to do ice and heat alternating. If my bf is home from work i usually ask if he will massage my hands and forearms. Makes a huge difference. Helps with blood flow