This topic can get confusing because some companies label their stuff stabilizer while others call it final rinse. The need for a traditional stabilizer to prevent dyes from fading is no longer needed. However, you should be still using is some sort of final rinse with a anti-microbial in it. This will help prevent mold/fungus from destroying your film if stored in not so perfect conditions. Properly processed color film has zero silver left so it lacks the natural anti-microbial protection that b&w film has.
If you're getting water marks from your current kit stabilizer you can either add a few drops of wetting agent to that mix (most powder kit stabilizers are only hexamine and have no wetting agent) or buy a proper final rinse like Kodak and use that. Kodak final rinse is nice because it doesn't have any hexamine, so one less chemical I have to worry about. In your situation I'd just mix up a liter or two in a big bowl and just work through all the strips, hang them and let them dry.
If your film is stored well, in a cool, dry environment, you shouldn't have too much of an issue just leaving them as is.
Personally, I like the cheap insurance so I use a proper final rinse.