Hope we've not gotten our OP
blackrat too confused with all the various comments/suggestions.
Here's my take. (doing this from memory so I could be wrong) First I'd want to know what the current cut-in/cut-out settings are for the pressure switch since you've messed with them.
* With power to the pressure switch and all faucets closed, using your pressure gauge on the schradder valve at the top of the tank, watch the reading and start bleeding the pressure slowly. Note what the pressure is in the tank when the switch closes. That is the cut-in pressure.
* Next, leave the pump running and note the pressure in the tank when the switch opens. That's the cut-out pressure.
With that knowledge you can make any adjustment necessary to set the pressure switch to what you want/need. Should be 2 adj screws on the switch. One sets the cut-out pressure and the other is the differential pressure, i.e. the pressure drop before the switch closes. I'd adj the cut-out pressure first then tackle the pressure differential adj screw. Each of these screws are compressing a spring. Cycle the tank a few time to see if the switch adjmt has been taken care of. It would be good to begin by dropping the pressure in the tank to insure the tank pressure is lower than the cut-in pressure. Note, the pressure differential should remain fairly consistent regardless of cut-out pressure. 20 psi is the norm.
* Next, with power off, open a faucet and let the tank empty.
* With the tank empty adj the pressure in the tank, as has been said a couple times, so it is several pounds less than the cut-in pressure.
* Cycle the tank a few times to insure the pump kicks in when it should.
You should now have things working right if all is OK other than the various pressure settings.
Now for a question. I see from the pic the water pressure gage is reading somewhere around 48 psi. Please explain that reading. Is that the initial reading, constant or max. If the If that is the max, you're going to have to adj the pressure switch to no more than that reading. One more question, how big is the new tank? Draw-down capacity and not the size of the tank itself. My tank is ~80 gal with a draw-down of ~25 gal. You want to size the tank so it doesn't cycle too often as that is hard on the pump.
Both adjmt screws on the pressure switch and the pre-charge pressure in the tank are interlinked and you may have to make some fine adj to get it all balanced out.
Hope I've not been too confusing. For sure I can do this a lot faster than writing this post.