I did a Google search and figured out the problem!
What was happening was that because the thermostat connector pins were so close to each other, when I last plugged in the thermostat, 2 pins (presumably the A/C and furnace control signals) were pinched together into the same pinhole connector and were shorted.
Everytime the A/C was turned on by the thermostat, I think it triggered the furnace as well, and so I had a mixture of hot and cool air, the room temperature thus never reaching the set temperature and making the A/C fight to stay on, trying to reach the seat temperature.
Anyway, I found the 2 pins shorted, (1 of them had got bent in the process). I unbent and straightened the pin and plugged in my thermostat and the system seems to be working fine now (It seemed to be cooling fine, with the vent producing cool air strictly). I left home for work this morning, but I will test it again, later this evening and also test the heating independent of cooling..... I'm fairly sure I have fixed the problem though....