IMHO if your trailer tires are 5 years old, it's time to change them regardless of what they look like. You are flirting with tire failure and it's cheaper to buy new tires than repair your trailer.
I know you check your tire pressure often. But, have you ever weighed you trailer? The pressure printed on the side of the tire is for operation under the maximum load. If your load is under the maximum, the tire pressure should be adjusted accordingly. Most tire manufacturers publish pressure to load tables for each brand of tire. The only way to know for sure if you are operating at the correct inflation is to know the weight that's actually on the tire.
That said, it could also be an alignment problem as already stated.