@kfitz It looks like you’ve found at least one option for archiving your Tweets on your own site, but do watch out for the archived t.co shortlinks which may not survive if Twitter were to disappear altogether. 
I suspect that some version of this option I’ve done before will work, though I haven’t tried updating it recently: https://boffosocko.com/2016/12/05/how-to-own-display-your-twitter-archive-on-your-website-in-under-10-minutes/
Our friend @jimgroom@social.ds106.us has recently written up some details that get around the t.co shortener problems: https://bavatuesdays.com/archiving-twitter/
I’ve also heard that @darius@friend.camp is working on something for a public release soon: https://friend.camp/@darius/109521972924049369. He may still be looking for beta testers if you’re interested.

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