These days, it seems like just about everyone has to check your credit, even prospective landlords. You and I know that just because you miss a few credit card payments that doesn't mean you'll skip out on your rent, but your next landlord believes your credit report over your word.
Some rental properties have strict requirements and won't let you move in with credit blemishes. Luckily, every rental property doesn't have as strict approval requirements. Some landlords, particularly individual landlords not property management companies, don't check credit at all. Read: How to Rent an Apartment With Bad Credit.
If you're already renting and just want to move to another location, you can always bite the bullet and stay where you are. Use the money you would have spent on your new apartment to pay off some of your debt.