Yeah, those should ease off once you bump up to a stronger dose. They tend to creep back closer to the next scheduled injection though, right before it's due again.
Dropping weight faster than that raises the odds of losing muscle along with it. On top of that, you're still barely into the therapeutic dosing range.
On the topic of storage cases, one option holds twelve pens as an example. Plenty of similar options exist out there, worth clearly marking each one though.