I was going to suggest we all (myself included re-read the OP) as it was indicating a shortage of trim, or elevator authority or both. I have no reason to doubt this, after all, our illustrious web wielder has been doing this flying bit plenty long enough to understand the concept of getting a wing slowed down, regardless of it's length or shape. Never the less Tango's post is solid, and yet it differs from what you describe in your initial post. As a general rule a ton of throttle is not out of the question, but how quick you shove it in there, and whether the situation warranted it will determine whether you are hanging on the prop, or squirting yourself across the LZ.... 'panic and brap' usually ends up in the latter...
But back to what really brought you here... If you think you are indeed short on trim or elevator it makes sense to investigate and rule the pertinent items before trying to modify your flying habits to suit an ill flying aero. In fact simply getting in a new airplane warrants a look at most of this stuff... My PA20/22 experience is far less than most of the 'guru's' here, but my understanding of the rag and tube Piper trim system is not lacking. The way I would direct it is from the most obvious to the least likely, and that list would go like this;
W&B / CG ) surely you've directed this already, but have you ruled out weight in the tail without seeing what it would do for your ground roll. We all know we gotta get the mass stopped, but I can't tell you how many airplanes I can actually get in shorter with a little weight in the back.
Rigging) Have you taken a smart level to it? If you put all the numbers on a piece of paper, find your HRL and zero your level to it, you can go through the entire plane in a matter of minutes. Correcting things on the other hand may take a bit. A fairly recent rebuild is no indicator of todays rigging BTW, these rag and tube things move and tweak as much as a living tree....
The stab yoke) they fly just fine with the yoke upside down, they just run out of trim too soon as the taller side of the diamond runs into the jackscrew tower. Incidentally, yours should be upside down when compared to a PA18, which would make it an easy mistake for a rebuilder who dabbles in both.
The rear stab inner tube) Piper did not put a zerk in the rear stab inner tube (I do on every rebuild, and there is even a hole where you would weld a high heat nut to accept it) as such it is incredibly common for the rear tube to be anywhere from sticky to fused solid from rust and dirt. Your stab will still move some, but you will not have full range trim.
And last, but still fairly common) rear section of longerons starting to 'sag' uphill. The more nose heavy, or the more power you have to use on landing, the more common this is going to be. The sequence goes like this; you fly a nose high approach that leaves you landing tailwheel first. Mostly gently, and not very often, so no big deal

sooner or later your tailwheel finds a boulder or you 'arrive' a little more abrupt, and those spindly 5/8" tubes begin to start drifting upwards. Whats really happening is the tail is twisting about the axis of the stab. The worse this gets the less flair you will have, meaning more power on the landings and more tailwheel first landings. This would be one of the first things I would check on the longer bodied Pipers, as every one I have been in to has been bent uphill at least some.
Anyways... investigating all of this is not a couple hours work and at a minimum puts your mind at ease as well as giving you a sense of direction to focus your energy.
Take care, Rob