It's not a pleasant prospect. There are a couple of things you and your spouse can do to give you at least a better handle on whether you have continuing exposure for debt. This is a helpful provision but difficult for the indemnitee to monitor. If the debt is from a credit card or other revolving charge account, the indemnitor should agree to make no further purchases on the account. The best course is to block further purchases on the account.
Indemnities are helpful but incomplete.
The best course by far is for the two of you to get disentangled as thoroughly as possible, as quickly as possible.