What will happen to purchased apps if I log out of my Google account
When you sign out of the account from which you purchased apps , they will stop working as licencing checks are carried out in the background. Thanks to Izzy who pointed out in comments that this valid only for apps having vending.CHECK_LICENSE active and running. Apparently , all purchased apps don't have this permission running
I have not personally tried this but you may like to give it a shot. Set up a family account . Instructions here. I don't know if you need different billing address / payment method
Another alternative is to add the new Google account to your phone and only use that for mail, Calendar etc. In this case the apps would continue to work but not be updated as explained in Google help