This is obviously a topic that has been covered in great detail, but my current issue is nuanced enough that (I hope) it justifies a renewed discussion.

I have a 2001 4.6 HSE with ~195,000 mi on it. I purchased the vehicle roughly 18 months ago with nearly new air springs and a known valve block issue. The neglect and abuse finally took its toll and the compressor seized due to overheating in DEC 14. (Shame on me, got what I deserved, etc...) I finally got around to addressing the problems recently. I rebuilt the valve block using a kit from BECM Doctor using Paul's rebuild guide and replaced the compressor. The system was as leak-free and smooth running as it's probably been in years.

Fast forward a couple weeks and the EAS is on the fritz again. The compressor is running almost constantly when the vehicle is in use and getting VERY hot. After some troubleshooting, here is what I've got:
1) Vehicle running normally - achieves all ride heights quickly and smoothly (according to lights in dash switch); vehicle makes a series of height adjustments/exhausts air at the end of every compressor cycle; compressor almost immediately begins running again
2) Vehicle running, door/tailgate open - compressor runs for a reasonable (minutes) amount of time and shuts off
3) Turning vehicle off - EAS makes a series of unnecessary lowering adjustments/exhausts air, but stops at a slightly lower level than set (e.g. standard to somewhere between standard and low)
4) Turning vehicle on - Dash lights indicate vehicle is lower than set (e.g. solid standard, blinking high lights) and immediately adjusts to level set when vehicle was turned off; compressor cycles on
5) Vehicle stays level and does not appear to get any lower after sitting undisturbed for extended periods (days)
6) No EAS faults (hard or soft...yet)

Item (2) tells me the compressor is fully capable of pressurizing the tank to the pressure sensor's satisfaction. I think items (4) & (5) tell me that the tank and springs are not leaking.

I really love the EAS (when it works) and want to preserve it if at all possible, but I'm nearing the end of my patience and budget. Does anyone have an idea what might be causing the current difficulties?

It is normal to hear an hiss of air as compressor stops. It is normal for EAS to level when ignition is turned off. Look in the "HOW TO" for my EAS sticky and go through the checks laid out in it. You may wish to check that all heights are set correctly and read live data from sensors then compare to stored.

If you have access to a Nanocom you can take a live trace and see how often it really is running. Anything up to 40% of the time is probably OK.

If it is leaking a lot then most likely cause is a leak somewhere. I hit a pothole at speed and it jarred one spring just enough to leak past the lid, so to speak.

You could fit Druims mod which consists of a simple LED light that lights up when comp runs, I had issues with my comp running to often , it was down to poor calibration on the front, got the front sorted and fitted Druims mod and now I can monitor the length of time the comp runs for which I am happy to say only for a couple of minutes on first start up and then cycles now an d then but only when town driving when you stop and start at lights or in traffic.