Sorry to hear about your troubles, it's never a good feeling when a problem suddenly becomes worse. Vector games are their own beasts when it comes to troubleshooting but perhaps I can shed some light on where to go next. I've fixed many Star Trek boards and I'm in the midst of a repair now so the schematics are pretty fresh in my mind.
The power supply outputs four voltages: -5 V, +5 V, -12 V, +12 V. The +5 V is used on every board, as this is a common voltage for almost every IC. The other three voltages are used to supply power for the op-amps on the speech and audio boards, the op-amps and DACs on the XY timing board and a few other miscellaneous ICs. From your description of the board swaps into Eliminator it sounds like the ROM and CPU swaps were a success and did no damage to anything in the Eliminator. These boards only use +5 V. The sound and speech boards, which had visible signs of damaged ICs, use all four voltages and seemed to cause an issue with the XY timing or control boards in Eliminator. Based on this, perhaps an IC on the sound or speech board (or both) failed in such a way that it created a short across the op-amps, DACs or other IC on the Eliminator XY timing board and took out those previously normally operating ICs. You do have a largely 'normal' looking picture so my gut says the DACs are okay. When they go bad, you get almost nothing close to a normal looking image. I've found many more bad op-amps than DACs in my repairs and op-amps are plentiful and inexpensive. The slanted text also looks like a timing/counter issue, which is controlled by some 74xx TTL but that can be saved for later troubleshooting once the boards are back to a stable state.
If I were you, here is how I would approach the problem to try to work your way back to a normally operating board set:
1) Replace any visibly damaged ICs or other passive components such as small capacitors or resistors. Look carefully on all boards since the voltage rails are common to all of them. Anything that looks like it took heat should be replaced.
2) With all boards out of the cage individually measure resistance to ground across all four voltage rails on all boards. This can be done by placing the black lead of your multimeter on the ground plane (the plane all around the edge of the board) and the red lead at the -5 V, +5 V, -12 V, +12 V at the edge connector pads. You should get roughly 100 ohms or more for +5 V and something in the kohms or Momhs range for the other three voltages. Anything close to 0 ohms or a very low resistance means there is a component on the board that is shorted to ground, perhaps from the power supply shock. Pull up the schematics and begin looking for ICs that use the shorted voltage. Remove one at a time and measure resistance after each removal. When you find the IC that results in a return to higher resistance that is likely a suspect IC.
3) Since you have new 2114 RAM on order, it doesn't hurt to replace them all just to prevent back and forth removal of the boards should any of them fail the RAM self-test. 2114 RAM is generally sensitive to transient voltage spikes and I've found dead 2114 RAM several times in the board sets I've troubleshot. You can always keep the originals as spares if some of them test good.
4) Replace all six boards in the card cage. Measure the resistance to ground similar to step 2 above, but of the entire card cage assembly. This can be done by measuring at the DC wiring harness edge connector on the side of the cage. Again, anything with 0 ohms or very close to ground should be considered suspect.
5) Double check all four DC voltages coming out of your power supply. Reversing the AC and DC could have also blown capacitors or other components inside the power supply. Rebuilding them is easy enough but it does take some time. Perhaps use the Eliminator power supply to test both board stacks until you can get back to the Star Trek.
6) Reassemble everything in the cabinet with the exception of the video signal going to the monitor. Any time you do repairs especially on the XY boards you risk sending some large voltages to the deflection board that could stress the big transistors on the heat sink should there still be an undetected or newly created fault in the X and Y signals coming from the timing board.
7) Power the game on, listen and look for issues such as smoked capacitors or other signs of power issues. If the game appears stable, give it a credit, see if it starts and plays blind (the sounds are good indicators of this). If the game won't boot press the red self-test button and see if the red LED produces any error codes for the RAM or halt signals. If the game appears to boot, measure the X and Y voltages at the video signal harness. Normal readings on AC volts using a digital multimeter should range from 0 VAC to 2-3 VAC and constantly fluctuating. Test for 30-60 seconds to get a good feel for range. Anything above this range and/or a static voltage that does not change indicates further issues with the XY timing or control boards. An oscilloscope is also a safe way to do this test but can be a pain to lug around. The multimeter method is usually sufficient to know you aren't going to overload the monitor.
8) If the video signals are nominal and the game appears to boot (no error LEDs) then hook up the monitor and see what you get. Run self-test again and cycle through all the screens. At this point, you should be really close to a functioning game. Any abnormalities in video or audio are probably only an IC or two away from fully working. If you still have the slanted text issue on Eliminator (or Star Trek) begin digging into the TTL that supports the timing of vector draws. It looks like an x-axis only issue so that can narrow your focus down to 50% of the timing board with only six or so TTL that control the x-axis.
I hope this helps!