I would definitely come up with some rules and like the idea of a cleaning fee so you can have someone come in and get the place ready for next time. Post the rules at the cottage, give a copy to everyone when they come to get the key.
I would add money to cover the basic supplies of toilet paper, paper towels, soap, dish soap, etc. as well as to replace anything that gets broken such as dishes, lost such as towels, and to cover some of the utilities. So if cleaning fee is $75, add $10 per night--7 nights would be $145, 3 nights would be $105. That is very reasonable--where else could you rent a beach cottage for that cost? Get the money--cash only--in advance. No money, no key.
I also suggest getting locks for some of the cabinets in the kitchen where you can store your personal food items. Same for a closet elsewhere so you can store your personal items--even if it means bringing in a new cabinet for that.
I would email everyone and tell them that as of April 1 (or whatever date you choose, but I would make it sooner rather than later), there are going to be some changes for the use of the cabin. Then state your requirements:
1. You must ask at least two weeks in advance.
2. Our family gets first dibs of course.
3. "No" means no, you will not argue.
4. There are new rules for using the place--list them: bring your own food, clean up after yourself, clean up the sand, etc.
5. New fees: cleaning fee of xxx and $y a night to cover common supplies, replacement of items.
6. Money must be paid in advance before the key is given. Cash only. (Do this or you will never see the money from some.)
7. If rules are not followed, people will not be able to use the cottage again or stay there with others who are using it.
You will have some people get upset with you, so be prepared. Do not JADE, just state calmly "It is our cottage and we want to have it for years. So, we need to be sure it is taken care of and unfortunately some people who have used it haven't so that is why we have developed the rules. If you don't wish to follow our rules that is fine, there are a lot of rental cottages available in the area."
Or do as the pp suggested and stop all use by others. Insurance issue is a good one to use and may be very true!