Schwinn "Anvils' are pretty much bombproof except for all the rust. The brakes should be pretty easy to service as long as none of the hardware is rusted together. Take the brake off and check the condition of the spring, it is steel and might have some rust on it, a little is ok but if it is seriously corroded it's toast. If it looks good take the brake apart and clean everything with a wire brush on a dremel. Then relube the moving bits and reasseble. Should work fine.

Should be easy to find some used brakes online, too. New ones might be more than you need unless you plan to restore the bike, might have some antique value some day(25 years)

edit: first check that the cable isnt rusted or pinched and that the problem is indeed with the brake. If the cable looks bad replace it.