I wasn't able to finish filing taxes, but I already entered my payment information and was told I paid. Is there any way I can get a refund for my payment?

No, but you can still file your taxes.  E-filing will be available through 10/15/19.  https://ttlc.intuit.com/replies/4232367

Hi, 

Unfortunately, I don't want to use Turbo Tax as there were a few tech glitches and I want to go with a professional, who would be able to help with multiple returns as opposed to just this one. Is there any way I can request my money, especially since I didn't file anything?

Here is the contact link for billing issues, but I believe the fee you pay is for tax prep and not filing, so I don't know if you can or not, but you can certainly inquire.  <a rel="nofollow" target="_blank" href="https://ttlc.intuit.com/questions/1899263-what-is-the-turbotax-phone-number?jump_to=answer_3300041">https://ttlc.intuit.com/questions/1899263-what-is-the-turbotax-phone-number?jump_to=answer_3300041</a>

Or maybe ask them to push you the download Desktop program.  It can do unlimited returns and efile 5.  Plus Desktop has more features and advantages over the online version.