The solution to the ODE $y'=y-x$ is $y=ce^x+x+1$, however, this is inconsistent with the slope field that I have created.
Consider the curve $y-x=C$. The slopes of the tangent lines of a solution along this curve are all $y'=(x+C)'=1$, and each point in the $x,y-$plane is on a line of the form $y=x+C$ for some $C$. Therefore, each slope in my direction field should be $1$, but this is inconsistent with the solution of the ODE. Where is my mistake?
If your equation was $y-x=C$ for some constant C, then you would be correct. However, in this case, $ y-x$ is the derivative(slope) at that point. For example, the slope at $(2,2)$ is $2-2=0\neq1$.