Yes OP screwed up by having an affair and losing touch with his kids but he can't undo the past. It's done. The OP can only seek to make amends (as much as he can) from now.
You have done wrong by your kids for 14 years, the longer you leave it the greater the wrong will be and the harder it will be to make amends.
Your kids may tell you to piss off but give them the chance to make that decision. As much as you can, without harassing anyone, try to ensure any message you get that says you are unwelcome comes from your kids and not someone claiming to know what your kids want.
If you don't have any other method of getting in touch then send a message via Facebook - to the kids directly if they are on it and if not to whoever put the photo up. Bear in mind that apparently messages will go to a separate folder marked 'other' which a lot of people don't look at if you are not friends on Facebook so no response could mean the recipient hasn't read it rather than not responding.
If you don't have one already then set up a FB account so you are searchable if they look for you.
There are organisations that can trace relatives for you but clearly that could be expensive.
If you do manage to get in touch with your kids and they tell you to piss off make it clear you would always welcome contact if they change your mind and make sure they have contact details.
Personally speaking if I had been in your kids situation I would be badly hurt that my dad hadn't stayed in touch but would at least give him a chance if he later got in touch. I would be further hurt if he had chosen to not get back in touch when he had the chance.
Good luck and I hope you get in touch and are able to re-build a relationship with them.