An app with full internet access does not mean it can log what you type. Only keyboards can do that.
That's why I NEVER use a keyboard with full internet permissions - that's the combination to avoid.
But I used to get confused when I saw the standard message that comes up when first enabling any new keyboard - it says the app can get my passwords, credit card info, etc. But that is a standard Google message for all keyboards. Only worry if it's combined with internet permission - otherwise it can't send it anywhere even if it did log my keystrokes.
Thank you for your response. I wish these weren't things people had to worry about with their phone.