For the SD cards, I suggest downloading the SD card formatter for this, it makes easy work of proper partitioning and formatting of the cards. I had used it extensively when I had my HxC going in my Emulator 2. https://www.sdcard.org/downloads/formatter_4/
I see you were active on the HxC forum, how far did you get through the debugging steps therein? Jumper settings, etc?
I'd start with the following. I know this is going to sound wee painful...
Find yourself a proper replacement floppy drive. Install that drive -- ensure that the system itself doesn't have a problem with the I/O bus. If you can actually read a floppy, format one, write to it, you know you're in good shape at least from the system board standpoint. Use both ribbon cables to test.
This guy has information on the actual floppy drive... http://floppy.shugart.free.fr/page.php? ... /index.htm
The reason I say this is because you bought a system with an unknown -- the bloody floppy I/O. It very well might be the floppy controller. Do you still have the broken floppy drive?
Meanwhile, the floppy cable you installed, if it's a simple 34 pin straight cable (with no intermediary connectors or twists in it) you should be able to connect pin 1 to pin 1 with a little bit of deduction on the system board and the emulator card. Pull it out, make sure pin one goes to pin one on both sides.
Ensure the jumper is set to ID0, drive A.
If you have access to, it would also be wise to test voltage of the connections to the floppy. I know from hard experience that a voltage fluctuation (or abnormally low voltage) to the emulator will cause chaos - but given the symptoms you have described, I'm less inclined to believe this would be an issue... but not bad to double check for any fluctuations or consistent voltage discrepancies.
I'd also download that SD formatter, format those cards using that tool. I suppose it's possible the formatting of those cards is dodgy as well, but I'm of the mindset you have something else underlying causing you grief.