No, don't cut them off whatever you do. If the main foliage further up is not healthy, that's probably a result of this last winter, possibly with damage to the trunk as well. The growth lower down is the plant's attempt at continuing life, and they should grow on healthily, taking over from the damaged areas above. Leave everything alone now, and next Spring, if the top has died back, cut it down to where there's healthy growth. Or if you can see the stem's rotting or soggy, cut it off, leaving that new growth.