How to estimate reaction products of competitive binding?
Why would you have to solve differential equations for this problem?  What algebra have you tried so far?  

If C is in excess, then there is no competition!  You should be able to model the reactions as just $\ce{A &lt;=&gt; AC}$ and $\ce{B &lt;=&gt; BC}$.  The equilibrium equation would be (for A) $K_a=\frac{ac}{A_{init}-ac}$, and you can apparently already calculate $K_a$ from the thermodynamic parameters, so that is an equation where the only unknown is $ac$, the amount of AC formed.  The equation is linear!

$ac=A_{init}\frac{K_a}{1+K_a}$

A similar result holds for B of course.  

Once you can calculate $ac$ and $bc$, you should be able to calculate the parameters you want to know very easily.

If C is not in excess, the equilibrium equations will not be linear and the balance on C will couple the two equilibria (competition happens!).