What adhesive should I use to repair my swivel headphones?

you would need to use the adhesive to maintain the mechanical freedom of the swivel. and contact adhesive or epoxy will work, so long as the mechanical alignment of the swivel has not adhered.

essentially, you can ONLY put the adhesive on the broken part.

I use a product called QuickSteel. It starts as a putty, you knead it and it becomes moldable. Mold it around the swivel part (however your headphones will no longer swivel but that is the price we pay for being Sony MDR-XB950 owners, they are easily broken.)

I permanently fixed mine to a permanent position. It kinda sucks but it works. I painted the dried QuickSteel with black paint and it matches the rest of the headphones however it’s not the most prettiest option.