as I understand it, sawed off shotguns are illegal in the US without a permit from the US Treasury, which doesn't issue them... making it unlikely anybody will get to carry one.
short barrelled shotguns & rifles fall under the category of "other" and the transfer tax is quite a bit less than the the tax on transfer of machineguns. Machineguns & short barrelled rifles and shotguns are not prohibited, only taxed. In the process of taxing you, the JBT's at ATF conduct a background check on you.