Not too far from you, in eastern PA. It's common on our hard water, which also usually has some iron in it, to experience the exact symptoms you describe on old well pump switches. The issue is not actually the switch in most cases, but a clog in the 4" x 1/8" or 1/4" nipple upon which the switch is mounted. I'll bet you a beer you're going to find yours clogged with minerals or iron.
Essentially, this restricts flow between the switch and manifold, and builds "lag time" into the switch. So, if water is running, you get the symptom you described, of the system running down to zero pressure, before enough pressure is let off the actual switch thru the restricted nipple, to register the low pressure and turn on the pump. You can ream out the old nipple, but for less than $5, it's a heck of a lot easier to just replace it. Likewise with the switch, there's likely blockage in the orifice leading into the switch diaphram, which could be dug out, but you have to be careful, and likely it should be replaced anyway.
So, when you pick up a new well pump switch, also pick up a new brass nipple upon which to mount the thing. Make sure to note your pressures (30 - 50 vs. 40 - 60 psi), and other features (ie. burst pipe trip), so you get the right switch.