NO, you cannot change the owner or rollover to an account in someone else's name. This is just like IRAs. However any funds in that account can be used to pay for medical expenses for your whole family regardless of what health plan you currently have. The restriction is when you can add money to the account, not when spending.
If your husband's employer is not paying for fees anymore, this can eat at the funds. It's worth it to find a low cost bank. Some are free with a minimum balance. If you have a low balance and can no longer add to it, it might be best just to use it up.