Alright, following up for anyone who finds this in the future. I soaked my rainfly in vinegar/water and then just ran it through the washing machine. I know that this is not good for the coating, but honestly I am just starting over with this rain fly. It was in such poor condition that I really had nothing to lose. The vinegar soak + wash removed almost 100% of the already flaking-off seam seal. Once it was dry I pitched my tent and put the rain fly over it inside out. I then resealed the seams with this product: Gear Aid Silnet Silicone Seam Sealer (available from REI and Amazon). I used an entire 1.5 oz tube and still need to do the seams at the ventilation area and the tent floor (I anticipated this and bought 2 tubes).
It looks great. I was a little sloppy with the brush so the new seals don't have the perfectly straight border line that you see on a factory sealed fly, but I am confident that the new seals will hold up. Here is a photo of one of the resealed areas:
Again, it ain't pretty, but it'll work.
After I set that in the garage last night to dry, I threw my foot print in the washing machine as well. I put that on the clothesline this morning then recoated the whole thing with this product: Gear Aid Seam Grip Tent Fabric Sealant (couldn't find it at REI). I don't have a photo of that handy, but it looks great as well.
The remaining work to do is to finish the seams, and re-coat the the inside of the rainfly and inside of the tent floor with the Tent Fabric Sealant product. Finally I'll spray the outside of the rain fly with scotch guard so the water beads up like it used to.
Overall I'm in about $50 on the materials to repair and a few hours of work. I really battled with the idea of just scrapping this old tent rather than sink money into it, but I'm glad that chose the latter. I am very happy with the results so far and look forward to having a "like-new" 13 year old tent. It would be a shame to see it end up in a landfill, although, admittedly, that approach would have saved me some time.
I've learned a lot from all of this, and will definitely be cleaning, drying, and storing my tents properly now that I've put all the work into undoing the damage due to my own laziness.
I can keep posting updates here. If anyone is interested please let me know. Otherwise just assume I did everything I said I would and that it worked out.