Check the statute of limitations for your state concerning credit cards. Maybe the statute runs out before 5 years, if so, you are no longer legally liable to pay the debts. You still OWE the debts, however, the creditor can no longer sue you to recover their money. Additionally, regardless of SOL, you may send the collection agencies a “cease and desist” letter or a “debt validation” letter.
A C&D will prohibit the collectors from calling or writing you, a DV letter will force the creditors to verify the debt. If the collection agency cannot verify the debt, they must remove it from your credit report. If the collector verifies the debt (legitimately), it will fall off your credit report after 7.5 years from “date of last activity”. DOLA represents the last time you paid the debt in payment or in partial payment.
Good luck, and keep the faith, you’ll get your mortgage sooner than you think.