Brakes don't need to be super-close to the rim for effective braking. It's not so important how close they are to the rim, as how far you are depressing the levers to stop. You should be depressing them far enough that your hand is in a comfortable place for applying pressure, but nowhere near bottoming out against the handlebar. Having the pads hit the rim with the lever about 1/4 depressed is generally about right. If the brakes are rubbing, spin the wheel and observe the pads in relation to the rim. Either (1) the rim is out of true, (2) the calipers are off-center, or (3) the pads are really closer than they need to be and the rims are only a tiny bit out of true. For your description, that the pads are not rubbing and then they start rubbing again, I'd guess you have a caliper centering problem. You can probably center the calipers by hand and then you use the brakes a few times and they are rubbing again. There is no universal way to center brake calipers because every brand is different. Some have a small screw on one caliper arm that will adjust the centering. On others you have to loosen the nut that holds the caliper to the frame, center, and retighten. Sometimes you can hold the caliper position with a cone wrench or flat 10 mm on the front side of the frame anchor as you tighten. On some older brakes you need to bend the caliper spring with a hammer and punch. If the calipers seem to be adjusted well and you still aren't getting good braking power, it could be the pad compound or grimy rims. You can try sanding the pad surface and cleaning the rims.
Oil is good, grease is better.
2007 Peter Mooney w/S&S couplers/Terry Butterfly
1993 Bridgestone MB-3/Avocet O2 Air 40W
1980 Columbus Frame with 1970 Campy parts
1954 Raleigh 3-speed/Brooks B72