Some small campings can be fully booked, but I don't think it will be an issue to get around. Remember there are also pits in many of the leisure harbours. Here you have access to good bath and washing facilities and often a view of the sea.
Denmark does not have "almansret" like in Sweden, so you are not allowed to camp free as you like. But you can always sleep in your car, as long as you do not camp (having the roof up, leaving things outside the car, staying for several days - it is always up to judgement), and there is no sign against staying overnight of cause (some popular carparks have that). I have done a lot of overnight stays outside campsites, and it is doable, as long as you use common sense, respect others and find a remote spot. But remember, while in the car, sleeping or not - do not to drink alcohol while in public areas. You risk a fine for drunk driving …
Besides campsites and harbours, there are a many private (farmers) that offers pits on their property. That can also be a pleasant experience. There are some apps for that.
Regarding what you should see and do, there are so many things and it is up to own reference what you like. In the meet prospect, there is an inspiration for this small area (Rømø), but there are also links to i.e.
https://www.visitdenmark.com/denmark/tourist-frontpage - from there you can go further, and find that each area has its own webpage. On larger gas stations you also find walls with brochures for all kind of things to see and do.
