Chop,
Without wanting to doubt what you wrote, are you sure to had full rudder in? It can be a bit intimidating to apply FULL pedal. I think back to crosswing testing I was doing in a modified Cessna Caravan last winter, where I purposefully chose a runway which had the 90 degree crosswind, with a steady 26 knots. Once on the ground, it did take full and sustained application of pedal to keep it straight, but it did it, as advertised. I did leave some rubber behind doing it!
If you are actually getting full pedal in (and no brake on either wheel), and that will not overcome a drift in the other direction, you have a maintenance issue. Rudder trim will not be a factor, you can overpower it.
Make the most of the rudder itself. Once on the runway firmly, hold the nose light with stick back, to the point where you have full nose up control in. That will let the rudder do it's job, in spite of what the nosewheel might be trying to do. Once the stick back will no longer hold the nose light, the rudder is of little use either, so you're fully into the steering. The aircraft should track mostly straight without much effort.
Though Pipers are pretty straight forward for nosewheel steering maintenance, I have flown Cessnas with "maintenance fail" nosewheel rigging, which were quite alarming to steer.
To challenge yourself, go and take an hour of compotently trained tailwheel circuits, you might rethink pedal use after that!
Good luck, keep to the straight and narrow!