In DC, ground is ground. You don't want to mix em between powersupplies, but you should be ok hooking the DC ground to either one of those. Do NOTmix AC ground and DC ground though. AC ground is usually separated visually and grouped together with the AC neutral and hot. From your description, the connection labeled GND is AC ground.
Hooking up the new power supply shouldn't be too hard. Any connections going to +5v DC will go to that terminal, you can stack wires on top, or splice them together. Your DC ground wires will go to the DC ground connection. on the A/C side, you'll have a white wire which will go to one of the connections labeled AC, a black wire, which will go to the other AC connection, then a field ground, generally green, maybe green and yellow, going to the connection labeled GND.
At that point, make sure everything is correct, then power it up and hope nothing smokes.
Something to check before you order a new power supply though: trace the ac lines that went to this power supply back to the AC in of the machine (with the power off) and see if there's a secondary inline fuse that blew. If so, replace the fuse with the exact type that was in there (or whatever the specs in the manual tell you) and rewire your original to see if it works.