JL, you can pump the unit down and change the schraeder valve in most cases without losing more than an ounce or so of gas. This is assuming the leak is on one of the main service valves at the pipe entrance to the unit.
To pump the unit down means you close the liquid line valve, pull all the gas in the system back into the condensing unit, condense the majority of it to liquid in the condenser, and then close the suction line valve. At this point, you'd have only gas in the lines and the evaporator, at a pressure of just a few psi. With the service valves closed, the schrader valves are still open to the lines, but not to the condensing unit. You can then change out the valve core in a few seconds and the quantity of refrigerant that bleeds out during the change is minimal. The unit needs to be in the cooling cycle for pumpdown, and its ideal to do it on a cool, but not necessarily cold day.
The unit needs to be running. Close the liquid line valve (small one). You'll hear the sound of the unit change as the compressor pulls everything out of the system. Once the low side pressure drops sufficiently, the low pressure switch will cut the unit off. Quickly close the suction line valve and cut off the power to the unit at the disconnect. If you've got a gauge hooked to the service port on the suction side, you can manually engage the compressor contactor to pull the system pressure even lower prior to closing the valve, but always stop when you still have positive pressure. If you pull into a vacuum, you'll pull air into the system when you remove the valve core.
Now you're ready to change out the valve core.
When that's done, open the liquid line first to allow the system pressure to equalize. Then turn the unit power back on, and if the stat is still calling, the unit should restart.
In the meantime, pick up a hex brass cap for 1/4" OD flare fitting at the local refrigeration supply house and use it to cap the port with the leaking valve core. The knurled plastic or brass caps with o-rings are designed to keep dirt and water out of the service ports and they won't stop a gas leak completely. The hex flare cap will stop it. Last time I checked, a 30# drum of R-22 was going for about $185, so you don't want any of it getting away unnecessarily. Re checking for other potential leaks on the unit, quickest way to spot them is by the presence of an oily film in one area. The ref supply house will also have a small bottle of leak detecting soap bubbles that'll allow you to pinpoint leaks. Pretty cheap. A spray bottle filled with household ammonia will also work well to spot real small leaks. For some reason, a freon leak that soap bubbles won't hardly show up will make ammonia fizz like crazy. Another cheap diagnostic tool
And the thing to always check first when looking for a leak is to make sure all the schrader valve cores are tight. Next likely suspect is the refrigerant line connections to the outdoor and indoor units if they're flare or compression connections rather than hard soldered connections.