Congrats!
You're right that you don't want to burn bridges. You never know if someone from your current company will go on to work for the same company you got the job offer at. There is always that risk.
I hope your therapist situation works out. Are you afraid your new insurance might not cover your visits? If you are, definitely get a list of insurance companies that she accepts. Some insurance companies actually utilize United Healthcare for what they call "behavioral health." So, they basically contract out mental health insurance services to United Healthcare, and a lot of mental health services across the U.S. accept United. So you'll have to do some research on that, but just FYI.