I've already started using 1Password but I'm now considering closed some accounts and calling some of these services to ensure they never give out my information for password resets and such... Scary stuff...
You should be using two-factor authentication for all your AWS accounts, especially the 'root' account that's tied to your regular Amazon.com login (I've always though this is a bit odd).
You should also never use that root account and set up IAM accounts for yourself and any other user (which also use 2FA).
If the service you're using doesn't support it, ask them to implement it. If enough people did it..
I'd still prefer the TOTP approach though because it doesn't require any connectivity on the phone.
Pay-as-you-go phones are advisable to use for two factor verification, as they are affordable and could be used only for this purpose. Don't hand out the number and you've got a nice disposable tool for protecting your accounts.