RCinFLA is correct. With a tapped winding that you have, all of the other suggested configurations are wrong. You have confirmed this with explosions and smoke. [edit] My apologies to ericgibbs: Your half-wave solution is the same as I suggest - don't know how I missed it on the first reading. You are correct. [/edit]
You have two choices:
1) Use two totally independent windings. If you use the transformers that you've described, this means you need two of them. With two independent windings, you can use two independent bridges each with their own capacitor and hook them up as you like.
2) Continue to call your 0V terminal "0V". Develop each of the two necessary voltages with a half wave rectifier (single diode) and capacitor. This isn't a good idea at the current levels involved.
In either case, you should use properly rated diodes. 1N4007s are only good for about 1A. You should consider 25A bridges for 10A; if your co-workers or local merchant don't know some, come back and we'll try to help.