10,800 BTU unit will blow but not cool air what could be the problem

It could be either the t-stat, the capacitor, or the compressor. It could also be a burned wire. You have to use a meter to test the individual parts to find out what is going on. DrTom4444

Can I recharge the unit myself

There is no need to recharge a sealed unit unless it was damaged and then you need to repair the Freon line that was broken. There is no indication that this is the case; it is most likely an electrical problem. You cannot recharge a unit without an education in HVAC. DrTom4444

If you can hear the compressor start and it isn't getting cool, the charge is probably low or completely gone. You don't indicate an age of the unit. It needs to be recharged by a LICENSED refrigeration tech.

If the charge is gone there is a leak which must be repaired. True, if the compressor runs but it is not cooling it can be the charge, but it can also be that the valves are bad in the compressor which results from the unit not being cleaned yearly. DrTom4444

There are times when a freon charge dissipates. Not necessarily a large leak. Auto A/C units do this quite often when the weather is extremely hot... Have it service by an appliance tech...