I was going to buy some brass and I wanted to know which is preferred, the Remington brass or the Winchester brass?

I prefer Winchester but RP is fine as well.  Whatever you buy, check the brass before loading it.  Many years (30+) ago, I got a box of new RP cases (components, not ammo)  where the flash holes were not finished properly; 50% had a thin layer of brass completely covering the flash hole & the rest nearly covering them.  All that was needed was to run a flash hole deburring tool through them.\ to clean them out


The barrel is pretty fouled with layers of carbon and my cleaning has uncovered some mild to moderate pitting about a foot below the muzzle.  Not sure if this should push me toward lead loads or stay with copper jacketing.  I do know you should strip all the copper out before going with cast. 

The fouling may not be copper.  Try soaking the bore for a few days with Ed's Red (available form Brownells or make your own - do a web search for the recipe)  Ed's Red is a superior solvent that dissolves and manner of crud.  I have a M1898 on order from the CMP with a barrel as you describe & will let it soak for a time before brushing it out.  I've used Ed's Red on a M1903 barrel that was clogged with debris from firing blank ammo.  It took several weeks as I changed the fluid 3x but it finally got it out.  At the end of the first week, a gray sludge poured out; the second week's drainage was thinner and the third week's fluid still was pink.