A/C not cooling to set temp, please help?!?

I suspect the unit itself isn't working right. 78 is well within reach for most systems even in hotter regions.

Go outside and feel the air coming off the outdooor unit while it is running. If the air is blowing straight up in a solid column of air, good. But if it spreads out in a cone (move your hand around about three feet above the fan and off to the sides) or worse, sprays off the top of the unit in a flat disc, then your coils are blocked up and dirty. They can't shed the heat that is moved outside by the compressor. That's bad but it's easy to fix, all you need to do is clean your coils.

Look up "how to clean air conditioner coils" and find instructions on condenser coils. That will take you pretty far. I would give more and better instructions and have done so for earlier questions on this topic, but I just don't have the gumption to go into it this morning. Sorry.

Good luck with it.

It could be your thermostat OR it could be where it is located and your house has bad air flow. So right at the thermostat it stays a bit warmer than some other places. When you feel OK see what the thermostat says and just set it to that. If it doesn't turn off then it could be wiring or the thermostat or the relay that controls the AC (but probably not the last one).

It just might very well be your thermostat if the house is actually cooling down. If you decide the change the thermostat and you do not get the very same exact kind, then make sure that you write down which each wire went on the old one first before attempting the replace the thermostat. If you do not know what you are doing, then I strongly suggest you buy the very same kind that you already have so that you can't mess up on the wiring.

As long as it is cooling your house, you can turn it on and off as you need to.

How cold is the air coming out of your vents? how much air is coming out the vents? some more info is needed to correctly answer your question.