That 5 1/4 minute is longer than mine. I am not in position to power up the engine at this time and test it, but I've replaced most of the stuff that relates to the air system leading to the purge valve, short of the compressor. If it's pressured up, memory tells me more like a minute between pressure up, purge valve off, and pressure up again.
It sounds to me like you have an air leak, or a control issue separate from the purge valve. I would make sure you don't have any leaks to the air dryer, like in the hoses attached to it.
I replaced the purge valve, and it's a relatively cheap part. But you may have something else leaking. I would first go to the pressure regulator on top the air compressor and tap it gently with a hammer or wrench to make sure it's not partially stuck. It may be causing the unloader valves in the compressor to unload prematurely, or may be partially stuck. Another potential culprit is the quick release valve, which is mounted above the driveshaft, and may also be sticking, and providing a slow leak instead of closing an opening.
These air leaks or controllers are tricky. You think you have it fixed, and it happens again. My experience says start simple, and first check for leaks everywhere you can -- to the air dryer, in the hoses to the air tank, to the quick release valve, to the air bags, to the roto-chamber (parking brake actuator), and around it, and to the dump solenoids on the air bags. These are not easy to check with the engine running, but you can air it up, shut the engine off and see if the air gauge shows a steady leak. If you don't have that, then I would go to the control systems above and check them out with the engine running, but make sure I had jackstands under the coach, or blocks, drive the coach up on blocks, or put the jacks down, to make sure you don't have a big air leak while you're inspecting, to make sure the coach doesn't crush you.
You can also put air on the coach through the air chuck with an external air compressor to check for leaks while you jiggle the lines that attach to everything on the air system without the engine running.
These are just a few ideas. If you have continued problems, PM me and I'll give you a phone number you can talk to me and I'll help you as much as I can.
Former Owner, 2006 36MDDS
2010 Arctic Fox 22GK Travel Trailer
2007 Toyota Tundra