Second picture looks to be aiptasia. Remove the coral from the tank, remove the coral from the plug leaving the aiptasia where it is, or scraping off the zoas. Rinse the zoas in clean water and reattach them to a clean plug or rock and put that in an observation tank if you have one, or keep an eye out for reoccurance if you did not remove all the aiptasia tissue during the operation.
The 1st plug, can't be sure? If those squiggly lines are the same creatures as shown on the 2nd plug then yes that plug also has aiptasia, but much worse. I would toss the whole plug, or carefully remove the clean area of zoas and toss the remainder to rid the tank of the infestation.
If it is a different critter (it kind of looks like an algae?) then a different treatment may be required? Can't decide from that first photo. Removing the plug and taking a picture in white lighting (in a glass) might help to id that issue.