First of all, you need to clarify whether the front end of the swing arm moves sideways inside the frame. If that is the case, either the grease seals are missing, or you need to add shims.
If the front end/pivot area does not move sideways, but the aft end of the swing arm moves, something is worn out or badly manufactured. It could be the bronze bushings, the steel sleeves, or the pivot bolt. Or a combination of wear on all 3. I would disassemble, de-grease, and check where slack/play can be found. Ideally, everything should be a light slip fit without any significant play. If you find play between pivot bolt and sleeves, you need to measure the bolt to determine what is worn. Likewise with the steel sleeves vs bronze bushings. Since you in any case need to disassemble, it makes sense to clean and check everything thoroughly, instead of just installing new bushings and hope for the best.
Surely you have some basic tools, like spanners, sockets and a hammer? That is pretty much all you need to remove the swing arm from the frame. Then you need a long punch, or just a suitable piece of small diameter round bar or tubing, for removing the worn bushings from the swing arm. To fit new bushings, you need a piece of 1/2" to 3/4" all thread, 2 large washers and 2 nuts. Just use the all thread to pull the new bushings into the swing arm. Easy peasy.