This year I received a 1099-G for 2018. Do I have to amend my 2018 return now, or can I somehow just add this to my 2019 that I am filling out now? Thanks!

Ok ... look at that form carefully ... is it the 2018 refund you received in 2019 and being reported on a 2019 form ?   

If so then it is reported on the 2019 return  IF  you itemized last year ... follow the program entry screens. https://turbotax.intuit.com/tax-tips/unemployment/what-is-a-1099-g-tax-form/L8j3PjezG

A State Tax Refund is taxable if you itemized deductions on that prior year's federal return and took a deduction for state income taxes instead of the sale tax.  You got a deduction benefit for it so now you have to include it as income. If you took the standard deduction it is not taxable and you don't need to report it.

If you filed the Free Edition last year you took the Standard Deduction and the state tax refund is not taxable and you don't have to enter it.