No, I think absolutely not a problem of car weight!

Cars "wiggle" because missing of a good working three-point suspension on cars! This is not a problem that can be solved by a steadier fixing of screws. Trucks need a free movement for following also a bad trackage. However not both trucks need should be absolutely free swivel, one yes, the second in a light limited manner.
Look the pictures and you know what I mean (and what I do with all my cars).

One frame bolster is equipped with a washer for an increased position which fits the correct high of couplers and so this truck can swivel free over its center in all directions!

The second truck lies on two off-center points so that the truck can not swing sideward. This is a typical three-point constraint that avoids wiggle of cars.

Here yet a different solution for the truck with the sideward bracing - short square brass profiles are glued on car boolster and the truck received two nails (or small bolts) on right position.

Brian's observation: (yes I know it is an older posting, but one that has wide spread application, and particularly for newer members who may not be aware of it)