moving time external machine drive to new computer

Two pieces of advice:

1) Yes, you can and should use the Time Machine drive with the new mac. In fact, when you use it to import your content, the new mac should inherit that backup drive, and you should be able to keep your backup history.

2) When doing this, you should migrate ONLY the user accounts and then install any applications fresh from the App Store or their websites. Do not bring old cruft to your new mac. This is the most common cause of performance and other problems, as people bring over stuff from years past, designed for an older OS and a different architecture, and they don't even know was installed - and then, of course, blame Apple...

Assuming your new Mac is operating properly, what not reformat the external drive for TM and start fresh?

If the drive is several years old, it may be best to replace it ... maybe get a second dedicated disk for enhanced security.