Your well is 120 feet deep, but your water level may be much higher, say at 40 or 50 feet.
when your well was drilled, the driller should have run a "draw down" test, which shows how much the water level drops in the casing under a constant flow rate. This information is usually on file with the county or state.
say your draw down is 10 feet and the static water level is 50 feet. That means you could raise your pump up to almost the 60 foot level without running it dry. Of course you wouldn't want to do that as that would be cutting it too close. But, you could raise it to say the 80 or 90 foot level.
The bottom line is you need to know the static and draw down water levels to set your pump depth.
My gut feeling is you may be ok going another 10 feet up, but play it safe and get the water level information first.
One other thing, sometimes a sand screen added to the bottom of the well is the only solution. This is not cheap, they are made of stainless steel and run in the order of $600 to $1000 installed at the time of drilling the well.