I am not sure what to use to secure the gaskets. I have Black super weatherstrip and gasket adhesive from 3M, will that do the job?

RTV would probably work well too

Will it fully stick to the foam gaskets?

i would use a spray adhesive 3m sells it.

I think it would you could always try one let it cure and see if the gasket stays put.

That would be perfect.

I used the 3M spray adhesive. I also had a lot of trouble getting the box back together with the thick foam gaskets I bought from CJPony. I ended up removing the 2 triangle foam pieces on the flapper sides and used homemade cork gaskets to make it thinner to seat better. I also lined one side of removable rectangle metal piece with cork as well.
The box went back together so much easier and doesn't rattle.

I have the CJ one too. My triangle parts are a little bigger to begin with and I thought with the thickness I would only be able to get one in. The unitas it was orig, didn't have those in place. I do have thin cork, maybe I will do 1 of each. I was on the fence with the gasket sealant as it seems thick. I used some rust pant after I did the initial wire brush so I could have a good surface to stick to.