I filed my federal tax return by mail in March, but never got any refund. I just got my refund from last year's and want to try to file electronically. How do I do this?

Once you filed your return by mail, the IRS does not want you to also efile your return. They request that you not do that. 

They just started opening the mail in June that's been stacking up. So you have to keep waiting.

See this IRS page for the latest information

https://www.irs.gov/newsroom/irs-operations-during-covid-19-mission-critical-functions-continue

Is it showing up at IRS Where's my refund yet?

https://www.irs.gov/refunds/

Make sure you are using the refund amount from your actual federal tax return and not the combined amount Turbo Tax showed you at the end.  That was for federal and state combined.

It is not showing up on IRS Where's my refund, and never has, which is why I am worried. I did not file state taxes.

Surely there's something I can do other than just not get my refund?

It will not show up on the Where's My Refund site until they start processing your return. They aren't processing mailed returns and haven't been for some time.