Textedit does not open text files.

Boot into Recovery (Command R) and from the dropdown menu: Utilities>  Disk Utility> run the First Aid on your Macintosh HD. If errors are found and repaired, run again until no errors reported.

macOS Recovery:

https://support.apple.com/en-us/HT204904

To trouble shoot further you can:

Try a SafeBoot https://support.apple.com/en-us/HT201262

Takes noticeable longer to get to the login screen, does a 5-15 minute disk repair before it fully boots up, and certain system caches get cleared and rebuilt, including dynamic loader cache, etc.

This test will tell you if third party interference; extensions etc are not loaded in safe boot mode.

Test issue in another user (or guest user) account  https://support.apple.com/en-us/HT204443

This will tell you if it a universal issue or isolated to your user/admin account. 

There is no standalone TextEdit.app you can refresh Mojave (and associated apps ie Textedit) from Recovery—

How to reinstall macOS from Recovery as well.

redwhale wrote:

I recently had to replace my internal drive, and I copied my previous drive onto it with Carbon Copy Cloner.

Something seems corrupt here if the text files are in the /Users/your-short-name path—in or under your home directory—and if you’re testing in a user without parental controls enabled.

...which might mean that the CCC clone might be the culprit. Maybe have a chat with the folks at Bombich, too.

Make an external backup or two, maybe a bootable installer, erase the storage device, and reformat as GPT with HFS+, reload macOS High Sierra, and migrate in your old content. Start over.

What specific version of macOS are you using per menu : About This Mac?

Ok. Certainly cannot reproduce what you describe here. Every application here that is capable of opening a text file (.txt) does so without an issue.

Pick a .txt icon and press option+command+i. In the Open with section of that panel, what is the default application set to open .txt files?

Try the free version of BBEdit...

https://www.barebones.com/products/bbedit/

Hmmm, any strange characters in the name like # or & ?

What rights are shown on that drive, maybe Rad Only?, & did you say it was HFS+ format?

Maybe uncheck ignore ownership on this volume?

Wouldn't think that matters, but something mysterious is going on.

What’s the path to the text file that you’re trying to open, here?

No futher suggestions that would not likely waste your time.