Log Drive getting filled up in SQL Server
I suggest you read this, as this is applicable in all scenario (even you have Always On Availability group) when your database is in full recovery model .

Why Does the Transaction Log Keep Growing or Run Out of Space?

Yes you should implement transaction log backup immediately.  Read this for details.

Best way to backup &amp; truncate transaction logs on a schedule

Shrink will not help as your logs are not being truncated. If you are truncating manually and shrinking you will not be able to recover point in time. Moreover files will grow again and you have to keep doing this manually all the time. Once you pre grow your file based on your usage and set up scheduled transaction log backup you should not have to adjust file size, truncate manually or shrink file.

After you set this up then you can decide if you need to expand your E drive.

I also suggest change your transaction log autogrowth from percent to a fixed size. Read this.