If connecting the load blows fuses, then maybe your load has a problem. What is your load, anyways?
Also, if you replaced a 18+18v transformer by a 35+35v, there can be problems. Over a resistive load, the current is the voltage divided by the resistance, and you doubled the voltage.
You can test the psu by connecting both grounds together. I would first connect the grounds at the transformer, with nothing else connected, just the transformer by itself, and chech that you have your 35+35v. Of course, before connecting the grounds, make sure that they're two independent secondaries, like you think. Measuring voltage over the wires of separate secondaries should give you no voltage at all. With the transformer unpowered you can also test resistance. It should be infinite. The transformer's datasheet should tell you this, anyways.
With the grounds connected at the transformer, you can connect the grounds at the output of your psu. Nothing should blow, or you still have a problem in your psu. Then you can test by using two power resistors as loads, from + to gnd and from gnd to -
Use resistors of enough power rating, and their resistance should be enough for the current to be below the limits of your psu.
I also think that using fast recovery diodes on a bridge rectifier is strange.