If you check from Rh/Rc to C you should only be getting 24 to 27 volts.
If you connect Rh/Rc to Y (bypassing thermostat) you should get it to cool.
If you're truly getting 60 volts something is wrong. What terminals where you checking across?
The control transformer in the furnace should only be putting out 24 volts. I'm wondering if it damaged your thermostat.
R is the 24 volt 'hot' lead from the transformer,usually colored red. From R to any other terminal should be 24 - 27 volts.
I wouldn't hook up another transformer until you figure out what's going on with that one. This is a pretty unusual problem.
Generally when a tranformer goes it goes open circuit and it's done. A high voltage is almost unheard of.
Double check your meter to make sure it's not doing somthing wacky. Then I would verify voltages right at the terminal strip in the furnace unit. R to C should only be 24 volts.
If it's more it's possible it's done damage to the control board also.