See my Sig on how to help out that filter of yours. It absolutely does WONDERS!
I struggled with that last year... this year, just don't have it up yet, what with planning for the new pool, being busy and all that.
Switching to sand this year though.
One thing my Sig doesn't say is I used the hoisery at Walmart in the little clear plastic eggs, jumbo size. They're like 33c/pair. I was able to reuse them for weeks, I'm just careful with them when I remove them to clean the filter, of which I also reuse and never replace.
BTW, everyone is advising you to just replace your water not so much because of the cost to shock. But the sheer difficulty in clearing it up with those little Intex filters, as well as the minor amount of water your pool holds compared to most.
So in the end it is cheaper and less hassle.
However, with that said, if it were my pool and after doing a drain and refill, I'd at least bring it up to shock levels and do an OCLT. Just to be sure there's nothing left living in the water that clung to the walls, or survived in pockets of water left behind when you drained it. You'll need to get CYA in the water too, this keeps the sun from burning off your CL. Do this before you start the shock process. See link to Pool School in my sig for info on all these procedures.
Edit:
Please fill in your signature and location info in your profile as instructed in the 1st link in pool school. This way we can better assist you based on your climate and equipment.