You cannot backwash shower filters.
There are other shower filters, but they will get clogged up as quickly as your current system.
It sounds like you most likely have a fair amount of sediment clogging up your filter.
You could get a filter for your whole house to reduce the sediment. It sounds like you might need this anyway to reduce some buildup on your plumbing.
Here are the housings for whole house filters:
http://www.waterfilters.net/Filters/wholehouse/wholehouse.htm
Generally, people get this size system for their housing.
http://www.waterfilters.net/culligan/wholehouse/housings/HD950.htm
Here is a good sediment filter for the housing.
http://www.waterfilters.net/culligan/wholehouse/replacement_filters/DGD-5005.htm
These last anywhere from one day to one year depending on the amount of sediment in your water. Based on what you are telling me, I'd guess that it would last 2-3 months between changes. However, this is just an educated guess, and your water quality and your usage of the water dictates how long it will last.
Thank you,
WaterFilters.NET