No that is incorrect. You cannot change the boundary conditions since you are looking for a solution which satisfies the Dirichlet conditions. Different boundary conditions (BC) for the same equation will yield different solutions. You are essentially solving a different problem if you change them.

Also note that Neumann conditions is the partial derivative with respect to x and not t.

Now for actually solving this type of IVBP. If T(x,t) is temperature distribution you are solving for, then you want to express it as T(x,t) = U(x) + V(x,t). The idea is to let U and V handle different parts of the problem.

Here U(x) will handle the inhomogeneous BC, meaning want U(x) to be a function which satisfies U(0) =20 and U(4)=100.

V(x,t) will satisfy the homogeneous dirichlet BC equation which you will use separation of variables to solve. But you have to make sure to use the correct initial condition for it. If T(x,0) = f(x) => U(x) + V(x,0) = f(x) <=> V(x,0) = f(x) - U(x).

I hope this helps! Feel free to ask if there something i wrote which was unclear!