It sounds like the chainring is "out of true" - that is, it doesn't stay in a plane while the crank is turning. There are multiple things that can cause this.

First, describing what you just noticed:
a) have you ever noticed this before? Or, are you confident it is a recent development?
b) does the chain rub against the front derailer while pedaling? The noise is an indicator you may have noticed before, even if you never noticed the "swaying" before.

Possible causes:
1) loose bottom bracket bearings. This is something that could have developed over time if the BB lockring came loose. (I'm pretty sure 1990 bikes still had manually-adjustable cup-and-cone bottom brackets). You can test this as the cause by lining up the crankarm with the chainstay or seat tube and trying to move it toward the bike and then pull away from the bike. See if there is any play.
2) the square-taper in the crankarm, or on the drive-side bottom bracket spindle, could have been produced a bit off-kilter. This could account for some lack of planing in the chainring.
3) the chainring itself could have taken a bump and gotten bent out of alignment at some point.