(OK, leaving aside the fact that there is absolutely nothing wrong with Scottish accents and I don't really see why anyone would be this keen to avoid them).
As far as I can tell, you really want to go to Glasgow. The only thing that's keeping you is that you seem to be obsessed with the following worst case scenario:
1. You arrive in Glasgow and everyone except you is Scottish / has already gone sufficiently native to speak with a Scottish accent.
2. Since all the people you talk to speak with a Scottish accent, you quickly pick up an accent yourself and after a few years that accent takes root and completely obliterates the English accent you had been cultivating up to this point.
3. The Scottish accent sticks with you even after you leave Glasgow. When you arrive back home, everybody thinks you talk funny, and you end up passing this funny accent on to your students.
What's far more likely to happen, though, is the following scenario:
1. You arrive in Glasgow and find that there are actually people from all over the UK - all over the world, even - and they all speak in a variety of different accents.
2. Since some (though by no means all) the people you talk to regularly are Scots and you're the sort of person who will quickly pick up accents, your own accent acquires a Scottish tinge, especially when you're talking to people who actually do speak with a Scottish accent. Not enough to be mistaken for a native Glaswegian or anything of the sort, but your intonation changes slightly, some of your vowel sounds start sounding a bit different and you pick up a few Scottish phrases etc.
3. After you leave, those Scottish features you've adopted ebb away, because people who pick up accents easily also tend to lose them again easily in new surroundings, as they pick up new accents. There might still be some faint leftover traces but nothing noticeable enough that anyone in your home country would pick up on it. It won't have any significant effect on your students' accents either.
In short: I really think you're worrying about nothing there. If you think the Glasgow course would suit you best, then go to Glasgow.