Yes, you can use your old Mac to file, but you cannot use the 2017 version of the software to file for this year. Each tax year stands alone and must be completed on form for that tax year. If you are using TurboTax, you must use the 2017 program to file a 2017 tax return and the 2018 program to file a 2018 tax return.
Here is a link to buy the 2018 program. You can choose between the CD/download version or the online version: https://turbotax.intuit.com/personal-taxes/cd-download/