Had a double break of my thumb last year after taking one on the hand...
You'll know if its a serious break, I don't think you would have been able to hold a bat. I felt sick and the pain was really very nasty! It took me ~7 or 8 weeks before I was playing again and tbh that was a bit early, you lose functionalitly through lack of use.
As for support, They can splint it with a metal rod in a mini-cast. If you still think its broken, I too reccomend A+E. They were swift with me. Even if they just send you for an x-ray and then confirm there's no break, its worth knowing. If it is a break, do not play. You'll certainly wreck your season and there's always the possibility of perma-damage! One thing you will notice with a broken thumb is exactly HOW much one uses their thumb in everyday life!