Air Conditioning - Central air wont come on

OK...this is my first house with central air in it. The problem I'm having is that it wont start when I switch the thermostat to "cool". The blower motor inside the house will run but I get nothing from the outside unit. If I hold in the contactor on the outside unit, it runs fine but only until I release it. I jumped the R (red) and Y (which on my particular unit is blue) wires on the thermostat and the outside unit will kick on and run fine (although I have to turn the blower motor to "on" as opposed to "auto" to circulate the cool air) I have the exact same results with 2 different Honeywell thermostats. One is an older dial type and one is the newer digital type which I installed late last fall and never checked it in "cool" mode until now. The dial type that I replaced did work the one time I tried it the day I bought the house, then I replaced it about 2 weeks later. I'm wondering if I have 2 bad thermostats on my hands or what other direction I might want to look at. Any help from you guys would be greatly appreciated as for the local HVAC guy just to pull in the driveway is about $100!

Do you have a "jumper" wire connecting R and Rc on your thermostat? If not then add one, check the operation and post back.

Yes. On the newer thermostat I do and it is connected between Rc and Rh. On the older thermostat, I have the red wire connected to terminal R (its the only R terminal...there is no Rc and Rh on the older one). I will try reconnecting the jumper just to make positive it is making contact at both terminals, although I'm thinking it is...