Brake and band clutch linings wear quickly, and tend to overheat easily. A readily available, woven, industrial brake lining may be the answer to your problems.
Sold under the name Redco Heavy Duty Woven Lining, it is available in a wide variety of widths and thicknesses, and for either dry or wet (running in oil) applications. It is not harsh on pressed steel drums, and lasts and brakes better than the old woven lining.
Any brake shop should be able to order it for you, as well as larger, full service, auto/truck parts houses.