Couple of possibilities.
Your alternator has nothing to do with it. Even if it is woefully deficient, power from the battery will still engage the a/c clutch.
Possible a/c maladies:
1. (easy) condenser (partially) blocked. To really clean the years of bug buildup, you need to get a very strong cleaning solution. AC supply houses have specialized stuff for this (condenser cleaner). Don't use the acidic ones. They work well, but unless you REALLY clean off the acid, it keeps on "cleaning" everything it has come in touch with. Remove the radiator hold down brackets so you can prop back the radiator, and back flush the condenser. Try this first. Even if this does not solve your problem. a clean condenser makes your a/c more efficient.
2. Since you just had it recharged, the charge level is suspect. If you have a leak, which probably prompted the recharge, the charge may have leaked out again. You have to ascertain (or have ascertained) that you have a proper charge and there are no leaks.
3. If 1 and 2 are passed, and you still have the problem, it would suggest not enough liguid freon is released into the evaporator. It working when you drive could explain this because the high pressure is higher when the engine is running at 2000-3000rpm versus idle.
This can be caused by a malfunctioning or partially blocked expansion valve (or txv). This is a fair amount of effort to get to (however it is much easier than changing a heater core).
The second possibility is a (partially) blocked high pressure line. The high pressure liguid freon goes through the drier which sometimes releases disseccant particles which will block lines when they leave the drier capsule. This you can diagnose by touching the high line all the way from the drier to the evaporator. The line should be hot all the way, and the same level of hot. If it goes from hot to warm, or even cool at any one point you have just found the blockage point. This you can fix by recovering all the freon first, disconnecting the high line at the evap (takes a 30mm wrench) and disconnect the high line at the drier, Now get an ac flush agent from an ac supply house. (you can get it in a can) and blow the line through in the reverse flow direction. Mount a new drier any time you open the system and put a new o-ring in any connection that you open. If you want the connection to seal for life, put some Nylog on the o-rings before you close up the joint.
Evacuate at least 3 hours, and leak test at vacuum to verify no leaks. Recharge to the proper amount do NOT overcharge as it will result in more horsepower drag from the compressor, and warmer air from your vents.