OP,
Sounds like you have a strong preference for the looks of the S197; if that's the case, then nothing you do is going to fix your Mach to your liking.
I've driven the Mach (my daughter's fiance has one), and the 05-09 S197. I own a 95 GT and a 13 GT500. I don't have the luxury of participating in HPDE, because I live in AK, and because I haven't put roll bars in my verts. So I have a little to talk from when I try to answer your questions about the comparative handling, but just remember this is my opinion, meaning your best bet would be to drive an S197 to form your own.
1) The SN95 can handle VERY well. My 95 GT corners like it's on rails; and the rear end will kick out when I tell it to, and only when I tell it to, and will do so in a very predictable manner. My suspension mods are pretty mild: Eibach Pro Kit (springs and struts/shocks), MM LCAs, subframe connectors, and CC kit with a good alignment, and good tires. If your Mach is that squirrely, I'd start by looking at the condition of what you had. Before the LCAs and subframes, my car was a lot looser, but now it is tight, predictable, and a joy to drive.
2) Shifter: Get an MGW. I've also used a Pro 5.0, nothing wrong with them, but the MGW is my preference of the two. An aftermarket shifter will make a huge difference in your shifting.
3) Overall driver feel: The SN95 platform feels smaller and more manueverable to me. Given two cars with equal acceleration and handling characteristics, I'd take an SN95 over an S197 because in my experience, the platform feels more nimble. All of that is SOTP, I don't have empirical data to back it up, but you don't seem to be doubting your Mach because of empirical data, but SOTP.
In the end it's up to you, and if you're issue is looks, then the decision is easy.
Not sure what you're looking at in the S197 as far as year/options, which will influence price, but Machs are going for around 12K in Alaska, S197 GTs start around 15k for the early models. You could do the mods listed above on your Mach for $1000-1500, and be in a great performing car for less than a used GT that will likely need some money spent on it to improve performance to an acceptable level for your road course expectations.