What does an individual's demand function describe?
The demand function does not say how much he's willing to pay for an additional amount dq. The demand function says: given the price of good is $p$, you will purchase $Q(p)$ of them. I think your confusion lies in the fact that it is possible to charge different prices for each unit of good. Notice that our theory is assuming one price, not multiple.

Moreover, the notion that consumer surplus measure the difference between what you paid and what you are willing to pay is completely false. Suppose the demand function is $Q(p)=-5p+100$. Suppose further that the price is $5$, so you would be purchasing at a quantity $75$. The consumer surplus is: $$CS=\int_0^5Q(p)dp=\int_0^5 -5p+100dp=562.5.$$ If interpreting it as the difference between what you paid and what you are willing to pay, we know that you are willing to pay $562.5+5\times75=937.5$ for $75$ units of them. In other words, you are willing to accept a price $937.5/75=12.5$. However, plugging this number into our demand function, we find that $Q(12.5)=37.5$, i.e. you are only willing to buy 37.5 units of the good, which clearly contradicts to our interpretation of consumer surplus.

For a better treatment of consumer surplus, I would recommend you to consult advanced texts on microeconomics theory such as Mas-Colell's or Varian's more advanced book.