The issues you will have will be the trim. We address this
two ways. The best (but of course the most expensive) is to
remove all of the trim, install the drywall, then install jamb
extensions at windows (if wood trimmed) and doors. Then
reinstall trim and base. This gives you a good long lasting
result that will last as long as any other similar wall system.

The other method is to leave the trim in place and run the
drywall up to it. Tape and float the drywall, then caulk the
drywall to trim. This is not as satisfactory because the
caulk may fail in places and have to be maintained. Also, you
must be much more precise with your drywall installation. In
addition, it takes most of the 3 dimensional look that your
trim provides, leaving you with a flatter wall detail.

A lot of what you do will depend on your budget and your
tastes. Some older (thin) trim may be difficult to remove and
reuse, so that may also be an issue. Alot of the older homes
had 1x4 or 1x6 trim and base, so if you have that in your
home, removing and reusing it is not usually an issue.

Ugh... You're right. Some rooms have new windows too, so I'm not
sure how much of the DIY job this is... For the rooms that have old
windows (which will be replaced eventually), would I be able to go the
caulk route temporarily, and then extend the drywall once the older
windows are removed?

Elana