Are you going to do the work, or do you have to pay someone to do it?

At that weight, I'll assume you have a low-end Nishiki from the early-mid-'70s with a frame of high-tensile steel.

Do your current wheels have aluminum rims? Are the spokes in good condition? If not, I'd recommend 'new' wheels, but you may have to buy them via Craigslist or a good LBS that's been around for a long while, since I haven't seen new 5-speed wheels in a long while (5-spd hubs are 120mm wide, 6/7 spd are 125mm wide, and 8-11 spd are 130mm (most road) or 135mm (some road, most mtn) wide.

If your rims are aluminum, I think your cheapest bet is to buy an old rear wheel in good condition. That way you can mount tires for one use on one rear wheel and for another use on the other. Next cheapest is to buy an old wheelset that allows even easier swapping, though probably at higher cost.

I don't know of an easy way to change handlebars.

What's your budget? $150-$200 should get you a used bike that has a frame built with plain-gauge 531 or even butted chrome-moly steel from Tange or Ishiwata and better components than your Nishiki. Especially since you want 2 different types of bars, 2 bikes may be your best option.