When parents divorce, it's not uncommon for their children to regress and start displaying behaviors they'd outgrown. Separation anxiety can be one of these behaviors.
Many divorced parents spend more time with their kids in the summer than during the school year -- particularly if they live some distance away and/or don't have primary custody of them. If you have a new significant other in your life, it may be difficult to keep that relationship from them -- and you may not want to. You love this person, and you're sure your kids will, too.