Priming one will be a pain in the posterior, after which it will probably work until it loses prime. I know, because I've done it (well - almost - not a well, but a large amount of water from about 8 feet down). I'd suggest buying a pump that will be less of a pain in the rear to use if you are going to spend money on this. I did have a free pool pump sitting there. They will not self-prime at all - It takes a large amount of water and a degree of persistence to get all the pump and hose full of water, put the output hose in a large buckefull of water, drop the inlet hose in the well and plug in the pump before it back-siphons all the water out of the bucket. If you have too many bubbles trapped, it won't go and you have to try again. Expect to spend at least half an hour fiddling with it to get it to start. If it's a long way down to the water, it gets more difficult to start or re-start.
A typical shallow well is sand-bottomed, so you'll be sucking sand through your pump. Not generally a good thing.
If you have a hill available, grab enough hose to run down it to a level below the bottom of the well, fill it with water, cap the ends, get one end into the well-water and the other to the point below the well, and uncap the ends - a siphon. Be sure to keep the well end submerged - it will move water until it sucks a significant quantity of air. With enough drop on the hill you can use a siphon to suck some sand or muck out of the bottom, too, with no pump to mind the debris. With a big enough hill you can put the end of the hose in the well, pull the lower end of the hose uphill a ways, fill just the lower end of the hose with water and then run it downhill, and the slug of water in the lower hose will pull water through the upper hose. If the water level to the highest point of the siphon is more than about 27 feet, you're out of luck for siphoning.
I would agree with @gregmacs comment to pull the pump up for service rather than pumping the well dry (if indeed, you can keep up with the inflow rate when pumping) - that is the way that sumbersible pumps are serviced. If your setup is missing the magical pipe joint (the pitless adapter - a cute little tapered wedged affair with an o-ring) as used on drilled wells, figure out a way to get one in place. If the pipe comes out the top, you don't need one, just pull the pipe up, pump and all.
You speak of digging the well deeper - how is the well cased/lined now? Does it have concrete well tiles in place? If not, digging it deeper can be a rather hazardous affair due to potential collapse. There are other hazards inherent in digging in a well (drowning, having buckets of dirt dropped on your head, gas pockets...), which is one reason that drilled wells are more common now.