AppleScript and TextEdit

For a start, nix the entire TextEdit thing. At best, it's completely unnecessary, at worst it's a huge bloat to your script.

Instead, since it seems that all you want is a resulting text file that contains the exiftool output. Since you have that in a variable TiffInfo, you can just write that to a file within your do shell script command:

do shell script "/usr/bin/exiftool -EXIF -Artist " & quoted form of POSIX path of eachFile & " > /path/to/output.txt"

This will write the output to /path/to/output.txt. Just change that path as appropriate, including using variable substitution to create a dynamic file name (other than saying you want each output in a different file you don't give any hint as to what those files should be called.

As written, this script will create a file called 'output.txt' with the result of the exiftool command. Any existing file will be deleted.

If you want to append the data to the file, use '>>' rather than '>'

If you want unique file names just build a unique filename variable (e.g. using some kind of counter, or based on the source file name, etc.) and use that instead of '/path/to/output.txt'

The options are almost limitless. It's up to you to decide what you want it to do.

What I am trying to get to work is if I have script run multiple time to have the output of each time the script run save to the same output.txt file.

That's exactly what the script should do right now. Are you saying that's not the case?