My mobile home has a laundry room ceiling light fixture that takes two of these bulbs: Sylvania 20671 - CF13DD/E/835/ECO Double Tube 4 Pin Base Compact Fluorescent Light Bulb.

It has worked fine for over 8 years without changing bulbs. We left it on 24 hours a day for 2 weeks while on vacation. When we came home, both bulbs were rapidly flickering. I bought two new bulbs, but the flickering is still there. If I leave only one bulb (whether new or old) in the fixture, the flickering goes away. Since replacing the bulbs didn't help, does this mean I have to replace the ballast, which is screwed/wired to the top side of the fixture? Where does one normally buy a replacement ballast? Or should I just get a normal fixture that takes real bulbs?

Sounds like the ballast, or the ground to the fixture is missing can cause the flickering.

An electrical distributor should have or be able to get you a ballast.

You may find it quicker and cheaper to change the fixture.