I have a '69 convertible. I don't like feeling like a 12' row boat on the ocean, the amount of body roll is insane. I think it's entirely stock. I want it to handle better. My intentions are to build it to be fast as heck in the quarter, so hard core cornering is not a concern, but it needs to be much better than it currently is (I will mostly use it as a daily driver). The full suspension upgrade kits are awfully expensive. What can I do to calm the body roll and bounce without doing all that? Can I just do springs, shocks, swaybars? Or is it like everything else in life; pay a ton and get what you want or pay a little and get nothing?

Can I just do springs, shocks, swaybars? why sure the factory sway bars and good shocks will help alot, on both the front and rear, new bushings on the factory control arms will help also.
you should also check to see that your frame is a Boxed convertable frame as when a regular chevelle frame is substituted the car will flex due to frame not being as stiff. this could cause handling problems:(. not saying anything but it has happened to guys before.:).

Good springs, shocks and sta-bars will help a bunch. Take a good look at the bushings and linkage, worn parts not only cause poor handling but are also dangerous.

I have meantioned it a few times here NYLATRON graphite inpregnated nylon no grain structure super tuff and more ridgid than polly-eurethane

you can buy it in solid bar form ..I bought a 4' x1.25" stick for 15 bucks from the local plastic suplier ... MAKE ENDLINK BUSHINGS out of it and replace the wimpy factory threaded rod with a larger grade 8 hardened bolt
You can cut the nylatron with a fine tooth chop saw or equivelant then taper the edges with a sander and drill the proper size hole Remove the endlinks at the drags for better weight transfer...

Once you start using these you will put them on all your vehicles .
my son and most of his friends are beleivers now