after down payment, you still have $10K, which should be enough to cover the
closing.
As long as the car loan doesn't show up in your credit score, it will not be
your liability, even you help your husband to pay.
But one thing you might need to concern that if in your account, lender can
see the regular payment to the car dealer, they might question you.
So I would suggest you that you pay the car loan from different account,
rather than the account you will show to the lender.