Are there bushings or some kind of sleeve in there that I can replace?
No. Old style hinge pins were unthreaded and held in place with nails, later styles were threaded. The pivot area is metal to metal with grease being the cusion between. The only replaceable pivot points on the transom assembly are the upper and lower needle bearings for the steering in the gimbal housing.
If there is that much play in the bellhousing/gimbal ring your only option is to replace whatever is worn. Could be the pins, the bellhousing, gimbal ring or all three.
Given the vintage your best option is most likely replacing the entire transom assy with a "new" one. Find a donor boat or ebay/c list for options. FYI it must be an mc-1 style, no alpha style gimbal housing will mate with the Ford flywheel cover