iPhone 6 processor destroyed due to swollen battry

Hi, did you try to replace the battery yourself and make sure that they weren't just telling you that so that you'd have to purchase a new phone / logic board. First of all, I would try replacing the battery and just making sure that they were not trying to con you whoever 'they' are. Second of all, if you have an iCloud account it should automatically back up your content to the cloud (if I'm not mistaken). Just sign into your new phone with the same iCloud details and it should work. If it does not, the only thing I could think of is that if your contacts are really THAT important, find someone who does board repairs and ask them to swap over your memory chips so that your old one with all your data will be re-balled and re-soldered onto your new iPhone logic board (I'm assuming that this must be done with the same model you previously had). You'd be talking a fair bit of money as board repairs are fairly costly. Hope this helps.