Sand oysters are no ordinary oyster. They're the most foul, cruel and bad tempered mollusks you ever set eyes on.
Anyhow, those midrange Weinmann sidepulls were sort of an exercise in mediocrity. If you keep them, I'd recommend using stiffer modern housing and cables, and upgrading to Koolstop or equivalent pads. If properly set up that should make them work well enough.
I'm not familiar enough with Schwinns to know if those were OEM or not, but sidepulls were the cool thing at the time, even for touring bikes. Weinmann centerpulls might feel a little stronger.
A nice thing about sidepulls that is rarely mentioned is that the brake pads stay in adjustment as they wear down. Practically speaking, this means you can get a lot more life out of a set of pads. Dual pivots - and CP to a lesser extent - have thinner pads that must be replaced more often for this reason.
If you are using fenders then definitely go for center pulls.