The nitrogen will dry the system, but not because it abosrbs the moisture or reacts with it.
Say you had a 1 cu.ft container filled with air at 0% RH. Then say you evaporate 1 oz of water into it. What you have is:
1 cu.ft of air with 1 oz. worth of water vapor in it
Now, start adding nitrogen. We start at 0 psig (14.7 psia). For every additional 14.7 psi the gauge reads, you have basically added another 1 cu.ft of gas (ideal gas law, assume that temperature is constant). Say we pump it up to 45 psig. This means we have:
1 cu.ft air + 3 cu.ft nitrogen + 1 oz water vapor
The water is still there, but is now more diluted
Now, nature likes to even things out. This means that the water is not going to stick to just the air, but will diffuse thoughout the space (as will the nitrogen).
Now, let's empty the system. The pressure is going to come down from 45 psig to 0 psig. There will be 1 cu.ft left at 0 psig, but guess what? What is left will probably be 0.75 cu.ft of nitrogen, 0.25 cu.ft of air, and 0.25 oz of water vapor. This is because everything in the container was in equilibrium and when you empty the system, you are removing a mixture (air, nitrogen, and water vapor) and not just removing the nitrogen or just the air by itself.
If you went to 90 psig, then you would have something like:
1 cu.ft air + 6 cu.ft nitrogen + 1 oz water vapor
So you can see how you can dilute the mositure as you add nitrogen to the system.
If you left the system open after you empty, it will then go back into equilbrium with the outside air (nitro leaves and replaced with air, mositure content equals outside air).
So, I can see triple evac could help. However, with the time spent filling with nitro several times, you could have been pulling your single deep vacuum during that time, and may be better off. In a large system or one that may have oil with mositure in the lineset, use of a triple evac may save time.