I wouldn't say synchro sleeve. 5th gear should have ring gear teeth, synchronizer, sleeve, gear, fork, and linkage. If it was a synchronizer sleeve (lol), then reverse would be affected as well, since reverse is the only straight gear on our transmissions. The rest are helical and 5th is sync'd with reverse on sleeve. just like 1-2, 3-4. I would check first linkage, then motor mount/trans mount.
2009 Cobalt LS XFE U74
Mods: K&N Typhoon Intake, Option Cat-Back Exhaust, Cosmo Short shifter, Maxxim Ahead 17X7's, MPx shorty antenna, FE5 Strut/shocks, Tein S Techs, Moog endlinks, Moog spring seaters, ZZP shorty header, LSJ downpipe, LNF front grille