Can I use turbo tax next year if I started to use it this year and then didn't file using Turbotax?

Well sorry but  you cannot have either option ... once paid the online fees are non refundable and the 2020 program will not complete a 2021 return. 

The only thing you can do is transfer the 2020 info you entered into next year when you start 2021 return.  So you might go into your account and make sure what is there is right.  And maybe tried to enter the rest of 2020 so it matches your filed return.