No the temp and fuel guages should work just with the main plug in the back.
If both sky rocket when you turn the ingition and/or when you only just get the engine warm it is probably the voltage regulator (pull the one out of your spare and give it a go) (the rectangular metal thing at the bottom of the dash pods)
If it is only one of the gauges that rises or doesn't rise then it might be a problem with either the sender in the block for temperatre or the fuel tank for fuel or it could be the gauges themselves. You can test the temperature sender unit by disconnecting it (it is near the front of the engine with a spade terminal on it) you then need to briefly short it out against the block if the temperature guage rises it is not the guage. Get a new temperature sender unit.
I had a dicky fuel guage for a while the gauges are easy enough to swap over so try the one out of your spare dash. (you said you had one right??)
Hope this helps
Oh and clean up all your electrical terminals I had a fuel guage stop working because the connections were dodgy I only really found out that was what it was when turned on the headlights and the temperature guage went through the roof
reconnected it and fiddled a bit and it was fine go figure
Most of the problems I have had with my mini have been in the dash