Yes, you can swim with the sock of stabilizer in there but in this post you mention that you are using chlorine tablets. Those are trichlor and they ARE stabilized -- meaning have cya in them. So, if you are going to use trichlor tabs all summer, you might want to nix the idea of cya in a sock or you may end up with too high of a cya level building up. Another option is to only add a little stabilizer to get to maybe a level of 20 and then use the tabs for awhile but monitor your cya level weekly. When it gets to about 40, you'll need to stop using the tabs and switch to bleach or liquid chlorine which won't increase the cya further.

You can shock to help with the cloudy water. Stabilizer won't do anything about cloudy water.

BTW - do NOT let the kids play with the chlorine tab floater.