I think a comprehensive check of the front suspension is in order.
For about $70, some parts suppliers offer a kit of some common replacement suspension parts. This would include tie rod ends, ball joints, and most front bushings. A great deal really.
My guess is that you could just replace the big lower control arm bushings ($10 each or less) and that will be half the problem. Additional play in the steering is often from INNER tie rod ends, the part screwed onto the ends of the steering rack.
These are not included in the kits and are more expensive, but not extremely hard to replace. Any shop capable of front end alignments should be able to replace these if worn. An alignment is definitely in order after all this work, and essential to getting the right "feel" in your car.
Also, as mentioned the rear trailing arm bushings, as well as torque rod and panhard rod bushings in the rear axle should be inspected and replaced as necessary.
After this kind of work, plus Bilstein Touring struts and shocks, and Turbo sway bars, my 87 240 is tight, solid, and handles extremely well.
Good luck with it.
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Rob Bareiss, New London CT ::: '87 244DL/M47- 234K, '82 245T/M46-182K, '89 244DL/AW70- 212K Not too distant past: 86 244DL 215K, 87 244DL 239K, 88 744GLE 233K, 88 244GL 147K, 91 244 183K

Four traditional methods for the normal person:

1) Polyurethane bushings all the way around
2) Upgraded stiffer shocks (Bilstein, Konis)
3) Thicker sway bar
4) Lowering springs

Heavy-duty sway bars are nice, but more importantly, you want to get ithe car into a muffler/brake/front end shop to have them check all suspension components. They will search for any problems with the hope that you'll pay to replace them. In my area, they inspect for no charge, you only pay for work. There are plenty of flexible joints and rubber bushings to check, and they're a lot easier to check on a lift than in a driveway.

If they want you to replace anything on both sides of the car, make them tell you which side is worse. Usually only one tie rod end or whatever is shot, the other side is OK. Many of these jobs are fine to do in a driveway; you'll make your own judgement on which jobs to do yourself.

If they don't find anything definitely wrong, go to a Volvo specialist. They should know every item to check. Or you could go there first, if you like.

I'll mention 3 items not often checked which can cause handling problems:
1) My '86 240 wagon developed loose u-joints in the steering column (I never learned the correct name for them, but they are a pair of flexible u-joints). Mine got loose, however they sometimes fail by getting stiff. Caused looseness in the steering. Volvo mechanics know how to replace with little hassle. Generic mechanics may think they need to disassemble the whole front end.
2) I'm told that failed rear trailing arm bushings will cause sway at highway speeds - in addition to a major clunk sound when putting into gear. This is about a $200 job in my area (Northeast US).
3) Failed steering rack. Hope you don't have that. If it's worn out, replacement is the only cure, at about $500-600 for parts+labor combined. Make sure you get a LONG warranty on the rack. Lifetime would be good.