Time Machine won't delete old files in order to have space to backup current data

Time Machine is software on your Mac that is used to back up the data on the Mac to another hard drive.

It sounds like what you are calling your Time Machine is really a Time Capsule.......a hardware device that is a WiFi router with a built in hard drive designed to store backups. Is that a possibility?

Open AirPort Utility on your Mac (Finder > Applications > Utilities > AirPort Utility)

Click on the Time Capsule to select it

Click Edit in the smaller window that appears

Click the Disks tab at the top of the next window

Click Erase Disk

Choose the Zero Out Data Option

Click Erase

The process will take some time........4-6 hours or more

When the process has completed, you can set up the new Mac to back up to the Time Capsule.

Time Machine can usually automatically delete enough older (and small sized) incremental files to make room for new (and small sizedl) incremental backups. But......when the backup disk down to limited available space.......and......a new larger than normal backup is attempted.......Time Machine simply cannot delete enough data to make room for the new much larger than normal backup.

Unfortunately, this is a detail that Apple neglects to mention.

You have two options at that point:

Option 1........erase the backup drive and start over again with a new backup of the new Mac and move forward.

Option 2......if you do not want to lose the old backups on the backup disk.......(few users ever need them).......you can use the Archive function in AirPort Utility to copy all of the data on the backup drive over to a USB drive connected to the Time Capsule. Once that is done, then you can erase the main backup drive and start over again with a new backup.

When you set up the new Mac to back up, Time Machine may display a message that the disk has changed. (It has, because it was erased). Click to accept the notification.