For those of you who know this problem well, I have a possible solution I need to vet.
QUESTION: Are we talking about a water pump priming problem?
If so, this problem does not seem to be dependent on the type of pump you have.
For example: In my RV I have a water drain line after the pump and when I do have a pump priming problem, which is very rare, I just open the drain ball valve and the pump primes... and then I close the ball valve 3 seconds later. Easy peasy!
So why did Winnebago eliminate the drain line?
My guess is that they started using a less expensive pump, gave you an adjustment screw no one really knows about, and they got rid of the drain line to save time and parts cost. That's what they do.
However, the RV owner on day in the far future, ends up wondering (often it would seem) how to prime their pump? So Winnebago's cost reductions once again cause RVers some frustrations.
INTERESTING STORY
In my case, for years I always considered the drain line after my pump; and the drain ball valve to be useless... until one day my pump just keep running... and I couldn't figure out how to fix it. Just like the OP.
All the faucets were off and there were no water leaks which would prevent water pressure from building up in the lines, sufficient to turn "off" the water pump.
Turns out my ShurFlow water pump had a air gap in it, just like the OP. (Albeit, we have different pumps, I know, but maybe this is the same problem. Is it?
In my case, the solution to priming the pump was this: All I had to do was open that drain-ball-valve and let water out of the drain.
...Then the pump would the pass the air bubble and after that the pump would prime. ...And after 5-10 seconds later water pressure in the line was achieved.
So what I am suggesting is this. If you have problems like the OP is having, is the best method to deal with pump priming problem... Just add a "T-fitting" to the water line coming out of the pump, and install a drain-ball-valve? ...Or is adjusting the pump pressure the right solution?
Note: I don't know if I'm confusing these two issues or not? After all we have different pumps.
However, I have never needed to adjust the screw on my water pump. ...And frankly I don't think my water pump has an adjustment screw, but maybe it does?
So it sounds to me that a better way to fix a pump priming problem is to just add a drain line to line-out of your pump, so you don't have to deal with this adjustment screw business, which I understand is sometimes hard to access. What do you all think? (Granted, I maybe off track so this time I need some commentary.)
I attached a picture of a Flait-it PEX type valve, but that would only work for if your water lines are PEX and you can install a drain line... by cutting a hole in your holding tank floor. ...In my case, I have a 3' flexible line I can snake outside when I need to, which is never. Or you don't even need a drain line if you don't mind water gushing out in your bay; and that too should not be a problem.