If we think about it logically though, there'll always be that O-H bond and it will always contribute to the intermolecular forces that bond alkanols together, no matter how dominant or not it is. Well at least as far as I know, lol.
You are exactly right.
What Stick and Ms smiley faces said are exactly what VCAA and your teacher wants to hear from you. These answers aren't incorrect, but they don't address your question directly.
The real reason is a bit more subtle than what has already been discussed. It's true that OH bonds will still contribute to the intermolecular cohesion of the molecules. It's also true that its effect will become negligible. This is for two reasons.
Firstly, hydroxyl groups interact with each other more strongly than the non-polar hydrocarbon bits. Some might call this hydrogen bonding (though it isn't, but that's a different can of worms). This favourable interaction gives rise to a strong cohesion that is much stronger than the non-polar to non-polar dispersion forces (or Van Der Waals forces). As the molecule gets larger, chances of two hydroxyl groups finding each other becomes smaller, so for the most part, these hydroxyl groups will instead interact with the non-polar groups. This means the extra cohesion from polar-polar attraction diminishes as your molecule gets larger. Therefore, as your molecule gets larger, the strength of cohesion coming from hydroxyl groups will decrease due to the sheer size of the molecule.
Secondly, the temperature difference doesn't scale precisely equal to the 'amount' of cohesion. As you get larger, more flexible molecules, other factors affect the boiling point, and it can be shown that the boiling point rises more slowly for larger molecules. That is, as the molecule gets large, there is diminishing returns for boiling point increases. This means the difference in boiling point will intrinsically become smaller. (for graphs, see figure 1
here as well as
this picture, note the curvature of the alkane lines.)
These two factors together means the difference in boiling point between alkane and alkanol becomes smaller as the molecule gets larger.