Weekly full backups and daily differential backups, saving the full backups from each week?
As you have stated, your backup strategy can be viewed as below :




  The reasoning for this type of plan is due to HDD size restrictions on the Azure environment I am working with.


Two imporatant things that you can do


Enable Instant file initialization (do follow other best practices when runing SQL Server on Azure VM)
Enable backup compression and take backups with WITH COMPRESSION



  Given that, is this a sound strategy, or where should I seek improvement?


From the above figure, there is a clear room for improvement. You should take a differential backup on Sat (just incase if a disaster happens). Also, its important to read Backups Gone Bad: The Danger of Differentials. Make sure if you have to take a full backup in the middle of your backup strategy, you use COPY_ONLY backup.


  Second, how do I go about retaining the previous week's full database backups but NOT the differentials? 


Dont use maintenance plans as they have noticeable drawbacks. I highly recommend to use Ola's backup solution (and Index maintenance  solution as well).

Edit : As per your comment:

If a full backup occurs in the middle, your differentials would be useless, since the base has changed and you dont know who took the last full backup. The differential relies on the last full backup. If you know people will take adhoc full backups that will mess up your differentials then deny them backup rights and create a stored procedure that will take full backup with COPY_ONLY option and grant those people execute rights to that SP. Additionally, do concentrate on recovery model of the databases - Full recovery model has to have a log backup.

You could even query msdb.dbo.backupset where is_copy_only = 0 to find out user_name of users who are taking COPY_ONLY backups

Just adding this for completeness - Restricting users to COPY ONLY LOG backups