As much as it hurts them to remember they work for you, I don't see anything wrong with a polite phone call at 4 weeks and again at 5 weeks if necessary. Good luck.
I went down twice before mine came in, and they were OK with it.
I called almost 3 MONTHS and the lady I spoke with was very nice and even explained 'why' it's taking so long. I mentioned that in the days before everything was on computers permits were issued on time. She said, yeah, don't I know it. BTW it came in a week later, 4 days before the 'grace period' ended.
Last time I renewed I just forgot about it and they mailed it to me. Of course my old PD was very LTC friendly. Before I left on the day I submitted my app. for renewal they wrote down my license number and told me what it would be when my license came in.
I have no idea what to expect here in this new to me town.