Hi silverman,
Welcome to Norton community.
Kindly let us know the flavor of N360 product you have (Regular or premier).
If it is premier then you can purchase a N360 multidevice key and simply activate as F4E 
suggested.
If the flavor is regular , kindly download the latest premier version from below link.
Norton 360 Premier - www.norton.com/latestn360p
Backup space allocated for N360 Multidevice and Premier is 25 GB . Backup space allocated for N360 Regular is 2 GB.
Product key purchased for regular will not work on Premier and Multidevice.
"will the new installation install without deleting existing settings in N360 such as logins details"
- Are you referring to ID Safe logins or Norton account login details that is present in product main UI ?
If it is ID Safe login , As a precaution I will recommend you to export your ID Safe data in both .csv and .dat format.
Thanks,
Lakshmanan