Would it change:
Feel
Stroke

Would it change the weight so much that it would seem like a whole new putter?

YES, it will change the feel, which may impact your stroke.

Shortening a putter (ESPECIALLY by 2 inches) would decrease the swing-weight of the putter, therefore making it feel lighter.

You would be taking approximately 20 grams off of the putter, so compensate you would either need to put 20 grams of lead tape on the putter, or put 20 grams worth of tungsten powder down the shaft.

I would recommend cutting off .5/1 inch at a time to see what you like best, and see how the weight works for you. But some people do like the lighter weight head and some like a heavier head.

When you cut it down, at first it will seem like you are missing everything, so give it a little time, don't jump to conclusions without a little time with it.

Good luck.