Both backup works independently. So, different HDD size is not a problem. I am currently running a 2T Time Capsule and a 3T Time Capsule.
As you can see, both TC have different HDD size. The bigger one has the backup up to July, and still has some free space. The smaller one only has the backup up to 2nd Sep, and is almost full.
If you simply install a 6T HDD and add that to the backup disk list. Then the Time Machine will automatically backup your Mac to either one of them. One hour later, it will backup everything to the other HDD. All your current backup still available in the old 3T HDD. And the 6T HDD only has the backup from when it install.
Since your 3T is too full and can't cleanup or backup anymore. Time Machine most likely will backup your Mac to the 6T HDD only but still leave the 3T HDD untouched.
So, you better copy all your current data to the new 6T HDD and then let the Time Machine to continue the backup.
What I recommend is:
1) Install the new 6T HDD.
2) Copy the old backup to this new HDD.
3) Add it to the backup HDD list in Time Machine.
4) Select "Backup now" to perform backup.
5) If backup successful, enter time machine to check if the old back is actually available.
6) Leave this config for few days, make sure the new 6T works well with all the backup.
7) Manually clean up the 3T HDD, or format it to start new backup.
8) From this point. You Time Machine will backup your Mac to the 3T HDD. One hour later, backup to the 6T HDD. And then back to the 3T HDD an hour later.....