In case others are also experiencing this problem, I encountered the same thing when I got my MODE recently and I think I've finally figured out how to fix it (after several false starts). At the very least this worked for me, and hopefully it will work for others as well. In short, the solution is: tape. Specifically, taping down the 21-pin cable to the metal plate covering the motherboard, pretty much everywhere the you can between the MODE and the socket on the Saturn motherboard.
When I first installed my MODE, after I booted it for the first time I noticed that the pitch of the audio during the boot sequence sounded odd, and even wavered a bit towards the end. I double-checked the connections (including the 21-pin cable) and while it seemed to get a bit better, it still sounded a bit off. The next day, however, it sounded considerably worse, with it being most notable with the opening cinematic of Shining Force III, where the audio was playing notably slower than normal. In fact, it wasn't just the audio, it seems that the entire opening cinematic played slower than normal, oddly enough.
I looked for solutions online and saw people mention that the audio pitch would waver when pressing on the 21-pin cable. I tried it, and sure enough, the pitch drastically changed by just pressing on the cable a tiny bit. Holding the cable down against the metal plate covering the made it sound better, and I noticed when I installed the MODE that there had been a piece of tape holding the cable against the metal plate, so I replaced the old, worn tape with a new piece, and that seemed to solve the problem.
Unfortunately after a few hours the audio started sounding strange again, even though the tape was still securely in place. So I looked for pictures online, figuring that perhaps the positioning of the tape and cable needed to be precise and attempted to match what I saw. I adjusted the positioning of the tape and cable, and again it seemed to fix it. Given the positioning of the cable, I thought perhaps it was an issue with the connection to the motherboard socket, but regardless it seemed to work... but only for about two hours. I tried it again, and the audio was again off. I tried putting another piece of tape on the cable, positioning it to ensure a good connection with the motherboard socket, which again seemed to fix it... but only for a couple of hours again.
At this point I was thinking that perhaps the cable was faulty. So I ordered a brand new replacement cable from DigiKey (part # 732-5099-ND, if anyone is interested). I replaced the original cable with the new one, without taping anything down just to see if it worked at all, and... the audio was now much HIGHER in pitch than normal. And once again, pressing on the cable caused the audio to drastically waver. In replacing it I also determined that the socket on the motherboard seemed to grip the cable quite well, dispelling my assumption of connection issues. Regardless, holding the cable against the metal place again seemed to fix the sound, so I folded the cable against the metal place the same way the original cable had been, taped it down, and it seemed to work... but only for about two hours. Came back, and it wasn't working again.
This time I decided that if it wasn't the socket, and it wasn't the cable, then it was just the positioning of the cable against the metal place. And thus I decided to just tape down every bit of the cable that was over that metal place between the MODE and the motherboard socket. And this time it seems to have held. It's been nearly two days and the audio has been working fine. I have no idea why this happens, or why taping it down fixes it, but it seems to work. There might be a specific "sweet spot" where you need to tape down the cable for it to work reliably, but every time I tried it would work for a couple of hours and then screw up again.
It's also likely that my original cable was fine, it was just a matter taping it down, but the replacement cable works fine as well, so that's what I'm using.