I found Authy unreliable, weird and not friendly... of course I'm a geek and and obsessive but still.
I use OTP Auth because is easy, clean, concise, has iCloud automatic backup and the most lovely feature is that you can add a widget in the notification, thus you don't need to fiddle with the phone for the common codes (basically the only I use like that is BW as others I generate them in the computer).
Recently switched away from Google Auth and yes: disabling 2FA and enabling with the new app is the way to go. Bit of a pain, but Google Auth has no recovery option (besides using your recovery codes for your accounts that is), which is annoying if you ever lose or wreck your phone.
I use both Authy (because I am lazy..) and OTP Auth. OTP Auth can be used for free (more options via in app purchase), is opensource and can export your accounts & sync via iCloud. iOS only though. And I see it has a Safari extension now as well. I bought the pro-version for a few bucks, no regrets.
You didn't state if you're using Android or IOS but for IOS I use OTP Auth specifically because it offers the ability to keep an external backup. I migrated from andOTP on Android to this and it was really simple to import my codes.