Back-up or start fresh with a new Mac?

I recently had issues with my MacBook Pro (late 2011) hard drive and had to have the genius bar assist with reformatting the drive. I then restored from my Seagate back-up (last "good" back up was March 23, 2018). I am running current version of High Sierra. Unfortunately, my Seagate will now not back-up my drive. It runs for hours to only say it can't complete. I have used Disk Utility to "repair" the Seagate drive twice. No luck. I also tried an older 1Click 1TB back-up drive that I had. It backed it up once, but won't do it again.

My question is this: I am now considering getting a new MacBook Air, but am not sure if I want to use the 3/23/18 Seagate ack-up to transfer to the new machine or just start brand new. I'm afraid it will transfer some "bad" files that are keeping the current drive from backing up everything. I do have all relevant documents (Word, Excel, PDF) that I want on a thumb drive. Thoughts? Should I just start "fresh" with a new computer and then only transfer over the documents that I want? Thanks in advance.