Most all weeping cherry trees are grafted onto other root stock these days (also called the stock root tree).* I suspect the sprouts are from the top part of the tree and the root stock has been damaged.* It can take a long time for the top to get some height, so you probably have a 5 year wait to see any height.* You might find these sites helpful to understand what I'm talking about.* This first site shows where the graft is made.
Do you know which species of weeping cherry you purchased?* This site lists several.
This site shows how to prune one that is grafted and limbs have sprouted from the stock root tree.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.