Can my boyfriend be held accountable for a debt that is mine through an oral agreement?

From the facts you have shared he could be sued, however because the burden of proof would be on the creditor and the statute of frauds would be instructive and be a good starting defense for both of you. You will need to consult with a lawyer locally.

Start keeping a detailed log of all calls and letters. You should also demand in writing all communications from creditors is in writing.

Do not give them any personal information because that is how collectors decide on which accounts to recommend suing.

If you are going to make payments use money orders and not personal checks or “check by phone” because if they find a bank account the collector will be more likely recommend a lawsuit the their legal department.

All collections are negotiable.
If you are going to settle mark the check “settled-in-full” at the very top back of the check and include a letter explaining you are offering a settlement, keep copies of everything

Get written confirmation of any payment plan the agency will accept.

I do not practice in your state and you will need to consult with a local lawyer for protection under your state law.

I have pasted a link to the FDPCA to help you with your federal rights;

http://www.ftc.gov/bcp/edu/pubs/consumer/credit/cre27.pdf

Good Luck