Can't transfer/delete pics and videos. Phone is full. Help!

My 5s is full, but when I plug into the computer to transfer and delete my pics and videos there are two problems (that started since the last update and that I have never had with any of my other iphones):
1. When I open the phone up in my computer, it doesn't see all the pics and videos. It only sees several hundred, but there are almost 3000
2. When I transfer the files I DO see to a folder on the computer and go to delete the transferred images from the phone, it says I do not have permission to do so.

I tried importing the photos, but the same problems happen with that approach as well.

Bottom line: I have always dragged and dropped all the pics and videos when my phone fills up (I have a four year old, so LOTS of stuff has been shot over the last few years!) and been able to delete from the phone afterward, but I cannot do those things now.

Also, my phone being full is preventing me from downloading the latest update.

I really appreciate the help.

- J