I'm being charged a $297.85 Tax offset from 2013 after using Turbo Tax to file my returns. This seems like a high number. on top of the $776 I paid in taxes that year.

You would have received some kind of notice from IRS concerning 2013 taxes sometime in late 2014 - mid 2015.  That notice would have explained what they changed or what you forgot to include on the 2013 return that made you have a balance due.

I was asked to resubmit my paper returns, which I filed with Turbotax, twice. But that was all.

Its hard to say without seeing the returns in question.  You can order a transcript of your account for free from IRS, it should show the figures that they based the taxes on, you could compare it to the figures you included on your return and see where the difference is.
You want the Tax Account Transcript.  https://www.irs.gov/individuals/get-transcript