carb problems?

The stalling out whenever you move the throttle position sounds like an accelerator pump. If the engine doesnt get that extra fuel at low revs when the throttle opens, it leans out and stalls, at high speeds momentum helps it along until the jets catch up to the new mixture. Still, check some of the easier quick fixes mentioned here first, either way, if the carb is old and hasnt been cleaned out in a while, couldnt hurt to crack it open take a look and do some cleaning while you're in there.