I, Carnak, will tell you your problem without even reading through RapidRobert's incredibly dense spewing wordforest.

Before you do anything else, find out what motor it is. Look at the front of the engine directly under the driver's side cylinderhead and find the engine ID number. Post that number here.

FORGET all talk of an internally balanced 360. If it had been done, someone would remember the $500+ balancing bill. It is NOT CHEAP to internally balance a 360 with stock internals. You need at least a couple slugs of Mallory and probably a lot more, and at $100 an inch it's EXPENSIVE! This is one case where it is definitely easier said than done.

Secondly, a 5.9Mag motor doesn't have a balance weight on the torque converter, it has the factory imbalance for the rear of the engine built into the flexplate.

Third, your picture shows a neutral balanced torque converter, never came behind a 360.

BUT, my "CHMSL third eye" tells me you have an LA 360. The "out-of-balance 360" post is probably the most common one in the whole history of Moparts.

It's possible that whatever doofus assembled the engine didn't know that the 318 damper and 360 damper were not the same. So check out that front damper and make sure it's for a 360. Clues are posted above. However, most engine shops don't have a pile of Mopar engine stuff lying around to mix up.

Once you have figured out that it's correct, and 2:1 it probably is, get an unbalanced flexplate (B&M is one source) for mounting a neutral balanced torque converter to an LA 360.

At this point your vibrating 360 problems should be over.

R. (Carnak)