To some extent your AC does have an integral role in some part of your cooling system hence when turned on your radiator fan kicks in to cool the engine down. Because the obvious states your air conditioning system uses power from the engine and can rank up the heat. This requires a couple of things to be checked, your electrical components leading to and from the AC to wherever connected as you mentioned you do not have a temperature gauge on your dashboard just a notification light when things get a bit spicy for you.
So let's talk a couple problems you may want to change or simply roll with the windows down as things can get a bit frustrating to fix. Especially with costs involved from a reliable and good mechanic/AC professional.
Condenser coils are prone failures over the years, your ECU can be going faulty and so to can your cooling system be going bad as in some vehicles they aren't as reliable. But the last or probably first thing one may do is perform a diagnostic test to see what is the code readings of your vehicle with or without the AC on.