I suspect your PSU circuit/wiring was off... Here's your 10-step program to happiness:
0) Take decent pics of everything and post them here
1) NO MORE POWER/ICs ON THE MODULEs UNTIL YOU FOUND THE PROBLEM
2) Remove all ICs
3) Disconnect the LCD
4) Disconnect the boards from each other
5) Power up the core and make measure all voltages. Post them here
6) Unplug the power
6) Connect the SID module to the Core module
7) Plug in the power to the Core modul and check for the 5V on the SID module
8) Add the power supply for the SID, measure voltages.
9) Report back with the results
For testing purposes(!) you can use that 12VDC wallwart IF and only IF it is regulated (ie delivers a constant perfect 12V). If that's the case you can simply wire it to the core's J1 for the core power (do not use the LCD with it for now because too much current draw will give you a pretty hot regulator), and also wire it directly to the output pins of the 7812 on the SID module (ground to pin 2, 12VDC to pin 3, see http://ucapps.de/mbhp/mbhp_sid_v3.pdf).