I would add another 5' joint of 1.25" galvanized to the shallow well and drive that point deeper, if it's just barely in the water table like that. A shallow well pump should be good for 25'. We usually do shallow wells at least "three joints and a point" here.
Tighten on a heavy-duty black coupling to drive with, and then remove it and install a new regular galvanized coupling over about 3 flat wraps of teflon tape wrapped clockwise only on the male threads of the pipe.
You need a backup wrench when installing couplings and pipe, and tighten it all as tight as you can. It shouldn't be able to turn when tightening anything.
If there are any leaks, you won't be able to prime it. Once you get it all air and water tight, connected to the pump, and have the check valve installed on the suction side, you can prime the pump.
To prime a pump properly, you need a gallon jug of drinking water, teflon tape, and a wrench or channel-locks. Open the largest 3/4" plug on top of the pump head, and wrap the plug threads with teflon tape clockwise only as the threaded end faces you.
Fill the pump head with water, promptly plug in or turn on the pump power while hand-tightening the plug. The water will bubble and sputter a bit as you tighten the plug. That's normal.
Hand-tighten it as much as you can and let the pump run until it picks up prime. Once it picks up prime then you can finish tightening it with a wrench.
If it doesn't pick up prime, repeat the priming steps, including the teflon tape.