The cover is off your pressure switch, which tells me it's probably had der fingers all over it.
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There are two adjustments on it, low pressure cut in and high pressure cut out. If you're screwed the cut out portion down (beyond what the old pump can give you)in an effort to raise water pressure, it will never shut off. Try backing off the high pressure (shut off) portion of the switch and see if it does shut off at some point.
You may possibly have a leak in the plastic pipe down in the well.
If I were you, I'd just pull the whole mess out and drop in a submersible and be done with it. Jet pumps suck.
ETA, most local codes require you to update the well (bring it to the surface) when you sell. (If the head is buried underground). If not the local health codes, then the lender usually requires it.