Air Conditioning question about Wiring versus Recharging

[jnf63]

The air conditioning in my 2g GSX does not work. The GSX had a 7 bolt to 6 bolt engine swap prior to the air conditioning not working. The air conditioning worked prior to the changes. My question relates to the possibility that since my driver’s side fan wire is no longer connected (see picture) that my problem lies there. Only the passenger’s side fan wire is connected to one large fan. Would the unconnected driver's side fan wire prevent my air conditioning system from working? I realize that it is possible that the air conditioning system lost its refrigerant while the engine swap was being performed, meaning that it probably needs recharged. Nevertheless, before I take my GSX to an air conditioning repair center for an inspection and a recharge, I wanted to understand about the unconnected fan wire’s possible relationship with my air conditioning system not working.
If I knew anyone with a stock 2g GSX locally, then I would simply have them disconnect the fan wire while the air conditioning is working to determine if the system stops, yet no one local has a stock system. Coincidentally, my A/C push button, on my dashboard, lights green, and my interior fan switch works, although my interior fan blows warm air.
Thanks for any help.

 

[Gus@RRE]

The fan being disconnected should not stop your ac system from working/ However without the 2nd fan you'll most likely start to overheat with the ac system on in stock and go traffic and warm days.

 

[jnf63]

The fan being disconnected should not stop your ac system from working. However without the 2nd fan you'll most likely start to overheat with the ac system on in stop and go traffic and warm days.

Thank you for the concise explanation.

I'll also utilize your advice about overheating issues, and I'll typically only use my A/C during longer drives, yet lower the windows while driving in traffic.