Water here is very hard. String filter is caked up after 3 weeks. Thinking backflush and wash with vinegar, reuse one time. The new filters are 4.5 inch and at replacing every 3 weeks, thats going to get expensive.
I think that you would be better served by a water conditioning company. When I moved into my new house 42 years ago I had a similar problem and I tried to correct the problem myself. When they failed I called "Culligan" who was the best company in my area at that time. A great gentleman came and tested my water. He recommended a solution that I thought was very expensive. I called a couple other companies but they were all reluctant to sell me anything. The guy from Culligan solved my problem. That man stated that my water condition was the worst that he had ever seen. 20 years I sold that house and the system still worked well with a minimum of service calls. I am glad that I called in a professional. my 2 cents