Is there a good reason to keep using FLAC v1.1.2 as recommended in Hydrogenaudio's wiki article on FLAC?
http://wiki.hydrogenaudio.org/index.php?title=FLAC

If v1.1.3 should now be the recommended version, I will begin using v1.1.3 with EAC to archive new CD's. However, should I bother converting the old FLAC files for any reason?

If so, I would appreciate a guide or some instructions on how to most easily convert the files with some verification of accuracy (before I delete the old files) and while retaining any embedded data.

No, 1.1.2 brings no additional feature nor security over 1.1.3. I changed it in the WIKI.
1.1.3 is more efficient (average bitrate is ~10 kbps lower). Converting from the previous version to the newest one will give you some free space back.