Yeah, there is that. Belding didn't seem to raise too much fuss about Alex having the gun, so a big ol broadsword strapped to the back might not be such a big deal. Unless FreakyBoy says otherwise, of course.
I'll just go ahead and have the sword in Adam's possession, but he won't actually use it until the points for it have been spent. If we need appropriate justification for why he doesn't think to carry it on his person, we can probably give that a handwave by explaining that Adam was under the effect of a localized idiot-ball generating raygun that Alex tested on him.