An alignment won't fix old worn out parts. I would start by replacing all 8 of the control arms starting with the lowers first. The arms will come with new bushings. After 15 years the rubber just disintegrates and the axle moves around. It's especially scary at highway speeds. After that you can look at your sway bar bushings, shocks if they are old, ect. The motor mounts and tranny mount are probably rotted as well.
After that I would get an alignment. A good honest shop will test your steering to see if the tie rod ends are worn out or not and tell you to get it fixed before they do the alignment. Doing an alignment on a vehicle with all that stuff worn out won't hold since it just moves around so much.
has some good prices. I used Moog stuff when I did mine. Even the cheapest Chinese crap arms with new bushings will make it feel better but they won't last and you will just waste your money doing it again soon. Better to get the higher quality ones even though they are more expensive. It's cheaper in the long run and less headache.
To answer your question, no having the rear sway links off won't hurt anything. Just makes it less stable on road.