Answer: First, under Arizona law a mortgage loan used to buy a home (or other real property) has priority over any judgments and liens against the buyer of the home. A.R.S. § 33-705. Therefore, even if you were the individual with judgments by the credit card companies, the title company should be able to issue a lender's title policy to the mortgage lender insuring that the mortgage loan is in first position on the home.
Second, the other title policy issued by a title company in a transaction is the owner's title policy that will be issued to you as the new owner of the home. A standard exclusion in an owner's title policy is that there is no coverage for acts "created, suffered or assumed" by the new owner of the home. This exclusion would eliminate title insurance coverage for judgments by a buyer's credit card companies.
Therefore, even if the title company assumes that you are the "bad" buyer with the judgments by the credit card companies, the title company should not be concerned about any liability either under the lender's title policy issued to the lender or under the owner's title policy issued to you as the new owner of the home.
Note: Even though this Arizona law does not apply to IRS tax liens which are governed by federal law, the IRS has ruled that mortgage loans to purchase a home or other real property will have priority over IRS tax liens. So even if this other individual has an outstanding IRS tax debt the title company should still be able to issue the lender's title policy.