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You can discharge your personal obligation for the community debt, but if you file alone, your husband will remain responsible for that debt. Unless he also files, the collection agency can still attempt to collect those bills from him.
By the way, debt that is incurred for the benefit of one spouse only is not considered to be community debt, unless the other spouse personally guaranteed that debt.
The creditor can get a judgment for the debt. This judgment can be used to garnish wages and bank account(s), and to place liens on property. However, because of the Homestead Exemption, which goes from $75,000 to $175,000 depending on the household composition, most homes are protected from liens, and cars with less than about $3000 - $3500 equity are also protected.