The 30-06 stock will fit your 300RUM action. A Remington LA is the same dimensionally regardless of what cartridge it may be chambered for. The only differences are in the feed lips and the magazine and the Ultra mag magazine is simply "cut out" for the larger case. If the feed lips were widerI could stuff my 338 RUM cartridges in my sons 30-06.

As far as accuracy goes,it is a shot in the dark. With a proper bedding job they could very well improve the accuracy of your rifles over the standard SPS plastic stockbut you never know until you try. You may have to experimentwith different loads again to find the one that it will like with the new stocks.

Apply several coats of a sealing finish to seal up the insideof the barrel channel and any other wood that is not already sealed or bedded to keep the wood from absorbing moisture. I absolutely love Permalyn finish, not only as a rock hard and beautiful stock finish but in the barrel channel as well.
http://www.brownells.com/aspx/ns/store/ProductDetail.aspx?p=7785&title=PERMALYN%20STO CK%20FINISH%20&%20SEALER

I say get the stocks. There were many a accurate rifles before plastic, kevlar, carbon, etc... were used to make rifle stocks. Wood is sooooooo much nicer than plastic on a rifle. Not that synthetic stocks don't have their place too.