Let's start off with the basics. For the park to work you need to 1) offer a terrific value 2) market that terrific value so you get 10 people a week wanting to move in 3) enact no pay/no stay to get those who can't work under the guidelines of paying rent to leave and be replaced by those that can.
When you talk about a weekly rental park, I immediately think of a park that is not very nice. If that's the case, you are attracting customers who are living there as a last resort, not because they want to. That's a bad position to be in, as you are going to have a revolving door of residents.
Figure out what you have to do to create an "insanely great value" -- the combination of quality and price. Don't shoot for just being cheap. You want to be of enough quality to attract residents who are employable and enjoy living there.
On the collections side, remember the old adage "it's easier to change people than to change people". You are trying to find residents who believe in paying their bills. Probably 50% of everyone who moves in might lack that skill. You can either re-train them through evictions, or let them move down the highway if they can't follow such basic life skills.
We have never had a park that could not have the collections problems tamed but ONLY when it is combined with making the community a nice, safe, clean, attractive place to live that the residents can be proud of.
I would also consider changing over to a monthly collections system instead of weekly. To me, weekly attracts the wrong sort of customer (typically people who are so irresponsible that they can't save up for rent, so they live life one day at a time). That alone may be scaring off decent customers, as it is very rare and a little alarming to most people.
What is the quality of your park and homes like? Are the other parks doing weekly rentals?