On a TR2-3B, the resistance should be nearly zero with the float at the bottom. On a TR4-6, resistance should be nearly zero with the float at the top.
Some digital meters do read '1' with the other digits blanked, to indicate out of range. However, the fuel sender on a TR2-3B should never go full open, and I think the same applies to the TR4-6. If it does, the sender is defective and probably will only work through part of the range.
My suggestion would be a new sender; but you've got nothing to lose but time (and sanity) trying to repair the old one. Open it up carefully, find the broken wire, clean the broken ends and twist them together securely. The wire will be Nichrome or similar, which is usually impossible to solder.
If you are really crazy, you can even remove the old winding and try to replace it. You'll need to find resistance wire with the same resistance per foot, and then duplicate the original winding.