What model Motobecane is this? Or, more important, is this one of the cheap ones where the dropouts are flat plate or a better model with a cast dropout? It is possible that a plate dropout can get so scarred up or bent that quick releases don't work very well. That will also be quite apparent looking at it.

Next question: does your skewer have a steel rod? Fancy titanium (or aluminum if made) will stretch more, exactly what you do not want. If your skewer isn't a modern quality steel model with internal cam (the lever coming out the side) get one. Shimano makes excellent skewers at all prices. A generic steel skewer from QBP will also work very well. (Modern skewers have better cam design than the old ones and are considerably less likely to open when not intended; a good reason to update if for no other reason.)

If you have a modern good steel skewer and the wheel still slips because the dropout is in poor shape and you want to keep riding the bike, replace the axle with a longer solid steel axle with track nuts. You will need a wrench from now on, but you will be able to ride in confidence. (My old Peugeot UO-8 went thousands of miles in that mode.)

As pratmann says, a too long axle is a show stopper. You can cut, grind or file it down (it is pretty hard, so it won't be as easy as it looks) or add small washers between the locknut and the cone. This will make the wheel a little harder to get in and out, but not much.

Ben