Somewhat late replying, but thanks very much for the suggestions. Oh and I have tried every cleaning supply in the house. I will try the baking soda, but do believe the poultice might be a good solution.

Carefully loosening the sand in the joints and shop vac'ing out. Time consuming but should be damage free.

So , regrouting after all is said and done is a reliable and sure backup plan. As well as it might look nicer to have everything all freshened up as well. Grout is cheap and easy too. ok that being said here is another option that I have often heard my painting buddies discuss but have never tried myself. Use petroleum jelly as masking over the grout. Brush it on the grout, do your refinishing, then clean it off. In the event you get some stain bleeding over( which I believe may happen no matter what) you can always go to plan B , regrout. If it were me I would plan on plan B. Petroleum jelly can be a contaminant for your finish and I think a good finish is more important than preserving some old grout that probably could use an update anyways

The results: Much better! I left a paste of Mt Dew and baking soda for about an hour, then scrubbed well with a stiff brush. Rinsed the patio off and let it dry--the stains are much better but not totally gone. I plan on using more Mt. Dew this afternoon, letting it sit for longer. BEFORE: (and this was after much scrubbing with Dawn and power washing!) AFTER: Thank you all for the great advice!! Even if it doesnt get any better than this, I can live with it, as there will be several flower pots in this area of the patio.