This formulation of the basis may be wrong, or I may be missing something, but I can't see a way to formulate the covectors this particular basis:
\begin{align} \vec{e}_0 &= \vec{x} + \vec{y} \tag{1} \\ \vec{e}_1 &= \vec{y} \tag{2} \end{align}
where a general vector in the space can be determined by:
$$ \vec{v} = a\vec{x}+b\vec{y} $$
I understand that the simplest basis for this space would clearly be:
$$ \vec{e}_0 = \vec{x} \\ \vec{e}_1 = \vec{y} $$
for which the covectors would be:
$$ \tilde{\omega}^0 = \frac{\partial}{\partial \vec{x}} \\ \tilde{\omega}^0 = \frac{\partial}{\partial \vec{y}} $$
where the derivatives are to be interpreted as:
$$ \frac{\partial}{\partial \vec{x}} = \left( \frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2}, \ldots \right) $$
In principle, the initial basis, (1) and (2) should have a corresponding basis in the dual space, since they span all the vectors and are linearly independent, but I cannot think of a function that would satisfy:
$$ \tilde{\omega}^\alpha(\vec{e}_\beta) = \delta^{\alpha}_{\beta}$$
(where $\delta$ is the Kronecker delta symbol)