The coil is either good or bad, only replace if required, the sleeves on the other hand do ware. See if the sleeve will come out of the bobbin, make sure the bobbin is not distorted where the sleeve inserts, check the Ω and if close the coil is good.
If you have the Schematic, the Ω is on their next to the coil symbol. Checking the Ω can be tricky to get a correct reading without removing. You need to have an open SW between you and the transformer and other coils. The transformer is easy, take the fuse out. On the previous page, that coil, Make sure it is the right coil, looks like a hack from another machine. At least it will be easy to check that coil,LOL.
Coils cost between $8 and $20 each, sleeves $1. There are a few types of sleeves; some extend each way some don’t. In my tool box, I have a bunch of LONG sleeves, I can cut then down as required, but it is nice to get the right one when possible.
Do you have a copy of your schematic in PDF?