Restoring from old hard drive

IF possible add the 2nd one to the 2nd connector of the notebook if you have. That would make it a lot quicker to do.

IF that is not possible you could add both to a desktop and do it from there.

Otherwise get an external casing (usb 3 connector at least because it will take some time), add the 2Tb to the casing. Then Attach the 2 Tb to the laptop

With those 2 disks available ...

diskdump the 1 Tb onto the 2Tb from a LIVE session to avoid changes to the active disk. After that switch the disks and use gparted to format and merge the 2nd Tb onto the 1st one.

Problem: Once I install the new drive and install Ubuntu, I would lose the environment, such as browswer cookies etc..

I would approach this in another way though that is more in line with a new install. You do THAT more often than replace a hard drive. And that method is repeatable with the use of a couple of scripts. Far quicker and more robust.

Install the new disk, install from fresh Ubuntu 20.04 or 21.04 (next lts) and then copy personal data over from the 1Tb (put that one in a case) to the 2Tb. Those browser cookies are not that interesting but you could copy them over from the 1Tb if needed. It is such a small file you could store it on google drive. That method is also possible with a lot of other settings.