Consumer preference and price in the Cobb-Douglas function
No, preferences are stable. That is not to say that the quantity demanded or marginal utility obtained at the new price level is the same though. 

If we'd allow the exponent of the utility function to vary for different price levels we'd let utility depend on prices, which it does not. Of course the actual utility that can be obtained depends on prices, but not the utility itself.

Back to your problem. You probably obtained a $\beta$ by observing one price and X quantity, with Y the remaining budget, filled that in in the formula: 

$\frac{dY}{dX}=-\frac{\beta Y}{(1-\beta) X}$

and solved for $\beta$, and then did that again for a different price and found a different $\beta$. Given that you use real world data, that in itself is not surprising.

Unfortunately, the real world does not always match our models, so you would have to estimate $\beta$ using econometrics. Given that in CD models expenditures are fixed proportions of the budget you should be able to simply estimate:

$\beta=\frac{p_x X}{M} + \epsilon$

That is $\beta$ is the average proportion of the budget spent on X.