I think that putting new shifters on old cables is false economy; cables are as much a part of the shifting system as the shifters and derailleurs and crappy cables have caused myriad problems. I also think replacing shifters which may be repairable, without attempting to fix them, is wasteful, but it is your money.
Note that the under-the-bar tape cable routing is much more likely to result in frayed inner shift wires due to the tighter bends, so check them frequently and at any sudden change in shift behavior.