700c covers a pretty wide range of wheels, too. are these vintage really skinny (and fragile) racing wheels? or 25mm wide (at the flats of the rims) 'hybrid' wheels? or something in between? what size tire are you using, and what PSI/kPA? how many spokes? How much approximately do you weigh?

for riding on bad roads, I'd want to use a 700x28 tire on a lightweight 'racing' type bike, and a 700x32 or x38 on a 'hybrid' or 'comfort' bike which can take them. I'd want 36H spoke wheels, with good quality stainless spokes. Cheap wheels are not very good quality, if I'm buying a replacement wheel, I'd expect to spend at /least/ $100 each, then spend another $20 or so per wheel having the bike shop fine tune them (I'm way out of practice on spoke tuning). Last pair of tires I bought were $50 apiece, plus inner tubes.

simply putting wheels on and off a bike is really easy, any cyclist should learn to remove a wheel and change a tire, fix a flat, this is really basic stuff. truing a wheel is a little more advance.

simple smooth side to side wobble should just be a matter of tuning the spokes, not replacing the wheel, unless the spokes are poor quality and corroded.