Why would my 50 amp legs read 130 on one side and 90 on the other?

Your read on it is right. When one leg goes high and the other goes low but the total across both still reads close to 240, that's the classic signature of a bad neutral. What's happening is the neutral point is drifting because the loads on the two legs aren't balanced anymore. The leg carrying more load gets dragged down, and the other one floats up. That's why your ACs are cutting out on the low leg and the microwave runs hot on the high leg. That high leg is what cooks electronics, so I'd get that site taken out of service til it's fixed. An RV plugged into 130 volts for very long can have a bad day. Now, where the problem actually is. The taped-up splices you found jammed off to the side are prime suspects. A corroded or loose neutral splice in the pedestal will do exactly this. So will a neutral lug that wasn't torqued down properly or has been sitting in dirt and moisture for years. Since your neighboring sites meter fine, you're probaly contained to that pedestal or its immediate feed, but not guaranteed. If both hot legs and the neutral run back to a shared feeder, a bad neutral connection at the panel or even a junction along the run can affect multiple sites, so neighbors reading normal is a good sign but not proof. Here's how I'd narrow it down. Meter the neutral to ground at the pedestal. On a healthy system that shold be near zero, maybe a volt or two. If you're seeing several volts or more, the neutral path has resistance somewhere. Then put a real load on it, like running an AC or a big appliance, and watch the leg-to-neutral readings swing. If the pedestal neutral is the problem, fixing the connections there will settle everythign down. If the swing doesn't change, start walking the neutral back toward the panel and check every splice and lug along the way.

One more thing since you're swapping that pedestal anyway. Kill the feeder and verify it's dead with your meter before you pull anything apart, because those crammed set screws and taped splices tell me somebody's been in there live before.

Ran into somethig similar at a buddy's seasonal park a few summers back, and the culprit turned out to be a splice buried in a junction box halfway down the feeder run, not at the pedestal at all. The connection looked fine at the box itself, but years of thermal cycling had loosened the wire nut enough that it only showed up under load. Metered fine with everything off, then the voltage would swing wildly the second a trailer started drawing amps. So if you fix the pedestal connections and the problem persists, don't assume the panel is clean either. Also, if any of those feeder conductors are aluminum, check for oxidation under the lugs, because that stuff creeps and adds resistance in ways copper never does.

Seen this exact pattern at the park I help maintain. One pedestal metered clean with no load, then dropped to 88 on one leg the second an AC compressor kicked in. Turned out the neutral lug had been over-torqued years back and the aluminum conductor had half cracked through inside it. Looked fine from outside, metered fine unloaded. So don't trust a no-load reading on that box. Put something with a compressor on the site and watch the legs while you wiggle the feed where it enters the pedestal. If the numbers jump around, the trouble is right there.

That second reply about the mid-run splice lines up with what I hit a couple years back. Had one pedestal doing the same swing trick and everything inside the box looked tight, ended up being water sitting in the conduit riser below the box. The neutral had corroded down at the bottom of the run where it was constantly damp. Pulled the conductors, they were green halfway up. Replaced the neutral and it's been solid since. So if the pedestal itself checks out clean under load, don't forget to peek at how that feeder enters the box.

Update: metered it under load. Dropped to about 84 on one leg the second a compressor kicked in, which sounds like what happened to that cracked aluminum lug in one of the replies. Everything inside my pedestal feels tight though, so now I'm leaning toward the mid-run splice theory. Walking the feeder tomorrow, checking junction boxes, breaker off first while I'm in there.