When a verb ends with a "e" that is pronounced, do you get rid of the "e" when you add "-ing"?
For example, would you say "His karaoking last night was really unique", or "His karaokeing last night was really unique"?
Unlike a case such as "make", in this case the -e represents a vowel in the pronunciation; that vowel remains pronounced when -ing is added, and so there's no reason to remove the -e. Hence: karaokeing.
As with a case such a ski > skiing, you may get a slight yod ([j] sound) introduced, but this isn't represented in the spelling.
You can probably just add -ing and retain the sounded
<e>. This would be the case for the verb recce, a clipping of recconoitre used in British English. I would write recceing. And in your case, karaokeing. In the past the hyphen was call into service for such situations (hence ski-ing) but we are less fond of hyphens these days.
For the past tense form, you can also add the -d the regular way. Again in the past, the apostrophe might have been used. Thus, karaoked (or possibly karaoke'd) and recced (or recce'd).