I put my 2 cents in on the barrel swap question you asked so I might as well get in in on the stock thing too!
IMO, the Accutrigger on the Savages really gets in the way of a good bedding job on the rear pillar. My BVSS stock had part of the factory rear pillar cut off (actually about 1/2 the diameter) so the trigger mechanism would clear. I ground a notch in the middle of the trigger mechanism right at the pillar location to give enough clearance for full support by the pillar but not enough to affect the Accutrigger engagement. I then installed a new rear pillar that actually supports the action. All that being said, I still like the laminate stocks better than the plain plastic stocks.
A guy at the range I frequent has a Choat stock. It works great! but it costs $221 at Midway. I seem to remember you saying something about a budget so I will assume that, like for me "money IS an object" and you can't just buy the best. I have my eye out for a lam stock for my 11 FCNS in 270 WSM but because it is a short magnum it takes the 4.4" hole spacing and is therefore hard to find.
A long action stock, on the other hand, is fairly common. My action is the center feed type and I have heard that the older stocks will not work. I will mention that I could take the bolt from my old 110 and use it in my 112 with no problems so I know the actions are THAT close in size. I would go for the action, barrel and stock in separate purchases. It gives you more options. You might have enough to buy an aftermarket Rifle Basics trigger which would eliminate the bedding issue mentioned above.
My old 110 action would have worked fine with my magnum face bolt and because the bolt face floats, unlike REM and company, you can get a workable setup for cheap. I used valve grinding compound to lap my bolt lugs on the 112 to silky smooth operation. Since you set the headspace yourself it is all "DIY".
My 7 Mag is shooting under 1 MOA at 100 yds with 110 grain Speer TNTs, RL22 at 62.3 grains with Winchester primers.
Here is a factory lam stock for a 116 which is the "new" stainless long action. At $189 it isn't bad.
Savage Arms Rifle Stock 116BSS Long Action Laminated Wood Brown - MidwayUSA
I paid $700 + FFL fees for my rifle new but that was a while ago and I have been shooting and not shopping! LOL I guess the big question is what kind of a deal you can get on an action...
That Choat stock is pretty good for the money.
Choate Tactical Rifle Stock Savage 110 Long Action Composite Black - MidwayUSA