Hi Josh,
The tire sidewall shows the maximum cold pressure - that is, max psi of 80 at 70 degrees F. Some people recommend reducing the tire pressure to match the tire manufacturer's inflation recommendation for the measured weight that each tire is carrying. That implies you have individual wheel weights. (The total weight on the axles divided by the number of wheels won't give you the real picture because trailer weight is not evenly distributed.) Other people recommend always inflating to the max cold pressure. 80psi in your case. Since most of our trailers are close to their max weight, usually the safest thing to do is go with the max cold pressure.
Once you start towing, the tires will typically increase pressure as they heat up. Sunlight also raises the pressure. That's why the recommendation is to inflate the tires when they're cool, out of the sun, and ambient temp is 70. It would be too complicated to figure out what inflation number to use after 4 hours of towing on a hot day in central Texas.
If your tires have been inflated to 60psi instead of 80, most manufacturers would probably say that you've been running on flat tires, causing irreversible damage. 20% low is considered flat.
Here's what happens. As you roll the tires down the road, the portion in contact with the road surface flexes because the weight of the trailer is pressing it against the immovable road surface. The flex produces heat. Under inflation produces excessive flex which results in excessive heat. Overloading does the same thing. Driving faster than the tire's speed rating of 65 does the same thing.
Tires are designed to shed a certain amount of heat that builds up. Once the heat buildup exceeds the tire's capability to shed heat, the internal fibers start to degrade and eventually break. Excessive flex/heat causes this damage.
Damage is cumulative and eventually the tire comes apart, usually while towing. There may be a sudden loss of air pressure which is quickly followed by the tire being shredded. The steel belts whip around causing damage to the trailer.
You should probably look at changing your tires. The unplanned expense will be upsetting, but not nearly as upsetting as the alternative.