1. I have a lot of debt. If you said this to me, I’d understand that you owe a lot of money. I wouldn’t know know if you have one account or ten accounts, but I’d know you owe a lot of money. 2. I have a lot of debts. If you said this to me, I’d also understand that you owe money to multiple accounts (house, car, four credit cards, etc.). Both sentences work, but I think you’ll find #1 works well in more situations.

also, "I am deeply in debt."