If you want some figure in the stocks, think long and hard before you cut. If I understand you correctly about it being a double stump, there could be some interesting grain at the junction. There is a grain pattern called "feather crotch", (not the name of a pole dancer in Witchita Falls), that occurs at the junction of a limb and the trunk that really looks cool in walnut. Again you can google that and see what I mean. Cut the rifle blank(s) first then worry about the two piece shotgun and pistol grips. If the rifle blanks don't come out right, you can always cut them up for your shotgun stocks.
You want to have interesting figure back towards the butt of the stock, and straight grain up in the barrel channel for a good rifle blank. Knots are a pain to work with around the action area or the barrel channel, so keep them behind the pistol grip.
As for drying, I think La Grande would be good. I wouldn't worry about the dryer. Just wax the ends really good, for about 4" up from the end, and give it good air exposure on all sides.
This is a stock I did once completely from a blank with only hand tools. I didn't want to screw up a nice blank, so this is a fairly plain piece of claro walnut. It takes a long time to do the inletting if you want a good fit. The Remington 700 is not too difficult to inlet depending on whether it is a BDL, or an ADL.
Once the wood is dry, you might think about sending it out for someone to semi-inlet on a pattern duplicator. It isn't that expensive, and they do all the hard work, and you get to do the fun stuff.
In any event, it will be at least a year or two before the wood is really ready.
Good luck.
Pete