Sorry everyone, I was literally searching another issue this many months later and noticed the responses on the post.
Just to clarify I have a 36gal primary tank, and a 5gal pig tank for expansion before the hot water heater. The 5 gal tank is right before my 50gal electric hot water tank.
So far the pump still sometimes double cycles in that 1/2 second and I'd suspect as it was suggested that the pressure is bouncing causing it to turn on and off really quick. I removed that check valve before as I was told that would cause issue like this, but it seems like I'd be better off with the check valve before the tank to prevent this?
Oh and 6 month update, pressure has been great. The only time we have issues is if the tank is low and we are using the sink and the washer is filling, maybe also if the toliet gets flushed at the same time. I've also had to stay on top of the filter chsnges but the about of silt has almost stopped. I'm no longer emptying out my first filter filled with silt every week. The last 4 months I only have to empty a 1/2 filled filter that happened all at once after we were gone from the house for about a week. It drew up a bunch of silt from it settling and then no more.
I have a feeling I'm going to need an impeller and other parts for this pump in a few years to get the best out of it. If this silt didn't put it to the test, nothing did.