That tyre is around 29" so you will reduce from 31" back down to 29" with what sounds like reduced offset.
Steering is affected by a number of measurements including caster angle. Changing the tyre and wheel size will change those measurements and angles and will have an effect upon self centering. However a 31" tyre is not a massive change so it may not be the total explanation.
If you want to get the centering back with 31" tyres you will have to have the geometry measured, work out what you need and get it changed. It sounds like you do not have enough caster, but best get it checked before going in and changing anything.
I am not sure what you mean by "loose". If there is excessive play in the steering, especially worn bushes, you will struggle to get a level of handling and self centring. If in doubt I suggest you get it checked and fixed before you spend any money.
The A/T will drive better than the M/T, they are more road biased. Not brilliant off road but you know that.
If your suspension is still correctly aligned for the factory tyres (a big iff on a 27 year old vehicle) you will notice a slight improvement when you change. I suggest strongly you have a proper 4 wheel alignment check before buying new tyres, you may find the results interesting.
BagusJeep lives in Bali.
1981 CJ7 258ci - Bagusjeep
1984 CJ7 258ci - Puthijeep
1981 J20 258ci - Gladys
1951 Willys CJ3A/MB/M38 - Little Willy
1995 Cherokee 4.0 - CHEROKEE