Is it better to file an extension if I don't have the full amount due? or is it better to file on April 15th with a partial payment?

If the return is complete, file the return, with our without any payment, just get it filed.

As Lisa995 notes, you should go ahead and file your return if it's complete, rather than file an extension.

But since you can't make the full payment right now, please see the TurboTax FAQ "What if I can't pay my taxes?" below for options.  A partial payment or installment plan, mentioned below, might work for you.

https://ttlc.intuit.com/replies/3301261