Using Time Machine to replace data on my hard drive.

You will find the guidance for restoring your browsers from your Time Machine backups here:

Restore items backed up with Time Machine on Mac - Apple Support

With the Time Machine backup drive connected to your Mac, you may either launch the Time Machine application or click the Time Machine icon in the menu bar > Browse Time Machine Backups and then navigate to the folder where your browser app is kept. That's usually the Applications folder. Then use the arrows at the right to scroll back to a previous date, select the browser and then click the Restore button.

Exactly what is it about your browser(s) that seems messed up?

The code portion of EVERY Application in the Applications folder does not modify itself -- that is a requirement for all well-behaved software on Apple devices. Restoring something that has not changed may be an interesting exercise, but is unlikely to solve the issues you are having.

Settings that modify the behavior of Applications are stored in your folders somewhere, and exactly where depends on what the issue is.