I teach clarinet to many, many kids of differing ages and hand size. This issue comes up all the time. For what it's worth, here are a few observations and possible cures.
1. You're new to clarinet and consequently, your hands are far more tense than necessary. The tension alone will give you soreness. I know you're coming from the sax and you might think you've learned to relax your hands but the clarinet is a different kettle of fish. In a few years, you'll look back and wonder how you ever played the clarinet with such stiff hands.
The cure here is time.
2. The angle at which you hold the clarinet.
Most beginners are guilty of looking down, dropping their head and consequently dropping the clarinet into a straight up and down position. Aside from ruining your tone, this puts all the weight of the clarinet on the side of your thumb where the thumb-rest contacts your thumb and bends your thumb away from your palm and fingers at an angle for which the human thumb was never designed to accommodate.
If you keep your head up and the clarinet out, most of the weight is taken not on the side of your thumb that contacts the thumb-rest, but on the "thumb-print" part of your thumb where it contacts the body of the clarinet. The higher your lift the clarinet, the more the weight is transferred from the side of the thumb to the thumb-print part of the thumb.
Our thumbs are designed to work in this "opposing the fingers way" and can comfortable support far greater weight than a clarinet.
The cure: Sit up straight.
3. Final thought. Instruments are designed by people who don't have to play them. If you hold your hand in front of you, as if you were holding an imaginary beer glass, you'll notice that your thumb naturally wants to point above your index finger at an angle. Your thumb points to 10 o'clock or thereabouts.That's one reason most people adjust the thumb-rest to its highest position. The problem is though that the thumbrests are designed in a way that forces your thumb into that uncomfortable 9 o'clock position. As others have mentioned, you can buy an obscenely expensive after-market thumbrest to counter this. Or, you can take 20 seconds of your time and simply bend the thumbrest in a clockwise direction until it approaches a 10 minutes to 4 o'clock position. I've done this safely on dozen of clarinets without a problem. On the older clarinets, you can unscrew the thumbrest first if you're worried about damaging the body. On the newer student models, just disassemble the thumbrest and then use a pair of padded pliers. This small modification has made a huge difference for literally scores of my kids.
The cure: Make the clarinet fit you rather than damaging yourself to fit the clarinet.
Best of luck.
