My first recommendation is to call Hornady. Their techs have ALWAYS been really helpful for me.
I got a Hornady LNL just a couple of weeks ago. Been handloading on a single stage for over 3 years.
A mentor warned me to take the press apart and clean EVERYTHING before getting started. Obviously I didn't remove the sub-plate or disassemble the linkages, but I did completely disassemble the priming system, clean it with rubbing alcohol and applied a light coat of REM DriLube to the contact surfaces. There was a warning somewhere about only tightening the screw on the part above the priming shuttle hand-tight or just a touch with an Allen wrench.
If the primer guide (that bent vertical tube) is popping out of the base, then something is catching on the primer slide inhibiting its movement toward the back. That's what the breakaway setup for the primer guide is for.
I can tell you that after about 300 rounds, the lower-front edge of the priming slide on my press was a little chewed up. I stoned off the rough edges and it seems to move more freely.
Another tip I read somewhere is that the priming system on the LNL AP requires rock solid mounting of the press. If the press mount isn't "bolted to the center of the earth", you may continue having problems. This isn't limited to Hornady progressive presses BTW.
If some cases are getting "torn up", it sounds like you are forcing things. When the press feels the slightest bit "wrong" STOP. Figure out what the problem is. I found that sometimes a case didn't get fully captured by the spring so it wasn't in exactly the right place for the sizing die.
Another example of this is that yesterday I was loading swaged Hornady HBWC for .38 Special. These are really accurate bullets but they're also covered with a lot of waxy lube which gets all over everything. No problem on a single stage press, but the gunk made the progressive limp along after a while. I started having problems with the spent primers not dropping completely out of the way. Stopping and reversing the shell plate just a touch usually resulted in hearing the primer and/or anvil drop, allowing me to move forward.
Also, be aware that if you don't push forward enough on the handle to fully seat a primer that will stop the press.
You should never need vise grips to insert or remove the LNL bushings. Something is WRONG there. My single stage press is Hornady too, so I've been using the bushings for several years. How much did you tighten the dies in the first place? When adjusting dies, you should raise the lockring and use your fingers to adjust the die height. Once you're there, tighten the lock ring down with your fingers. You usually do not use a tool to tighten or loosen the LNL bushings AFAIK unless it's just a little bit of snugging up.
I'd look at the bushings - do any of the engagement surfaces look chewed up? I'd definitely ask Hornady about this problem.
Powder Measure: A tip from a mentor is to run powdered graphite through the powder measure after cleaning everything. There's graphite in your smokeless powder as a lubricant anyway, so it won't hurt anything. Just run a bunch through the measure and leave it there. I mean a LOT.
I got really frustrated with my LNL measure a long time ago when I would get the charge just right then discover that after 20 or 30 cases it had drifted up a tenth or two or more. Adding a baffle in the hopper helped. Adding 1/3 of the powder, rapping the measure with a screwdriver 10 or more times, adding another 1/3, rapping some more, finish powder and rap some more pretty much eliminated the problem (except that the first charge in the drum always is high - just toss it). The powder settles and compresses a little as the measure jiggles every time you throw a charge. Even more so on a progressive press with everything happening at once. Now I've got a very small vibrating motor attached to the measure. Turn it on for a minute or two before weighing powder and I'm ready to go. Some guys use a small fish tank pump motor.
Finally, every powder measure is sensitive about how you throw the charge. You need to be consistent to get consistent charges. Yesterday, I experimented and confirmed that my charges where different when running cases through all 5 stations than they had been dropping a charge into a pan by using my hand on the measure. You need to dial the charge weight in while running the press the way you intend to.
BTW it's easier to get charges dialed in by weighing two charges (or 5 or even 10). That way you get an average and have a better idea of how much or how little to adjust the measure. I can tell you that the micrometer insert for the Hornady measure is fantastic. Results are very repeatable and the adjustment increments seem fairly linear (IOW if 13 tics on the micrometer made a 1/10th grain change before, it'll be really close next time too.)
Sorry - I didn't mean to type a book. There are a lot of things going on at once with a progressive press and it's easy to get frustrated when it doesn't work perfectly. I was really aggravated yesterday until I figured out that the bullet lube was the source of most of my difficulties. The clue was that the de-priming and priming had worked fine on my first two batches. That system needs to stay CLEAN to work right. Again, I doubt that's limited to Hornady's system. Oh, and loading powder into a case without a primer (we've all done it) gets powder in places you don't want it like under the shell plate and around the front of the primer slide.