First, see if they are the correct coils for your game (look in the manual).

New coils won't make the game play any better, so just replace the one that overheated.

New sleeves WILL make the game play better, especially if the old ones are worn.

Check the plungers, they should be smooth and shiny.

Check the plunger tips, they should not have a rolled, mushroomed tip.

Check the plunger links, they should not have too much play.

Check the springs, they should not be rusted or broken.

Clean the Switches by drawing a crisp dollar bill between the contacts.

Set the switch gaps, so they fire at the slightest touch. Bang on the playfield and make sure they don't fire from vibration alone.

Of course, replace the slingshot rubber while you are working on them, along with any cracked star posts.

Clean any black carbon dust from the coils, the underside of the plastics and the surrounding area.

Get ready, your rebuilt slings are going to hit hard!

Swollen coils happen, if it were me I'd just replace the swollen one. If the other slingshot coil still works and a coil sleeve is easy to insert, I'd just keep using that one - won't notice a difference. Bigger improvements would be noticed by replacing the coil sleeves (as you are doing) and lubricating the metal pivot point (I use Super Lube Teflon Oil, Pin Restore usually sells it for ~$5).

Oops, big delay in typing a posting, vid nailed it all!