Next to the .30-30 the 30-06 is probably one of the most popular deer cartridges in the US. As several former posts mention, it will do any job that needs to be done, with proper bullet / load selection.
Further, the .270 actually has some advantage over the 30-06 in the form of higher sectional density of the bullet for a given weight. That means that more energy will be retained downrange, for a bigger thump. The .270 is another popular North American game cartridge that was put in demand by the late Jack O'Connor. He took just about everything with it, including deer, sheep, elk, moose, and bear.
Your weapons may not be chambered for the latest sexy wonder cartridge, but there are tens of thousands of successful hunters each year that use the lowly 30-06 and .270. IMO, even though 350 yards is a stretch I would not trade it, and would use them with the proper loads.
If you plan a lot of 350 yard shots or expect to do a lot of mountain hunting in the US West, you might look at the 7MM or 8MM Rem Mag or the .300 Win Mag or .338 Win Mag.
BRS