Metal lamp sits on a cement floor. Its two prong and plugs into
A strip outlet by the computer that is three prong. If I'm barefooted and touch any part of the metal lamp I get a shock. I'm wondering if I should use a three prong adapter. If that would help??

The neutral wire is probably reversed with the hot wire, so that the screw shell is live instead of being the neutral, and the screw shell is shorting out to the frame of the lamp.

The neutral should be the larger blade on the plug, and should be the ribbed or striped wire on the 2 wire cord, look closely at the cord and you should se one wire is marker. This wire should be connected to the screw shell of the lamp socket.

I any case, the insulating material between the screw shell and the brass outer shell/housing is not insulating any longer and needs to be replaced, most likely.

Or could be the cord itself is just shorted out to the frame of the metal lamp.

The short answer is no a 3 wire adapter will not serve any purpose, and the lamp needs to be taken out of service and repaired, before someone gets hurt.

When I was a kid and played arcade games, I would sometimes get a shock, I would unplug and plug back in reversed. Solved the problem, Wouldn't be till years later I realized the cause(like yours), how reversing helped, and how to locate the problem, then fix.