I'm concerned that someone may steal the card and charge something, causing the balance to go negative for 'x' dollars, where 'x' may be many thousands.
Call your bank. I had a situation a number of years ago where someone deposited a fraudulent check in to my checking account then used a "dupe" of my debit card to run my account negative. It was a mess, but the bank made me whole. Your state/country may have laws related to your liability in these situations.
Would it be a good idea switch to a credit card with a low limit for the shared account?
Yes. I never, ever, ever, ever, use my debit card outside of the bank branch or bank owned ATMs. There is no point. With a credit card, if the card is stolen it's the issuing bank's money that's at risk, not my checking account.
Your issue isn't theft or fraud though. Your issue is you don't have a buffer for vendor errors. Credit cards give you a float period between when you pay a vendor and when you repay your bank. That float time is valuable because issues like the one you experienced can be flushed out and fixed before you experience any negative impact.
You may want to switch to two credit cards and pay them out of a common checking account. Switching from debit to credit won't change your fraud liability, really; because largely that's handled by the interchange network (Visa MasterCard whatever). But it would certainly give you the buffer to identify problems before your checking account is overdrawn.