Let $K$ be a finite extension of $\mathbb{Q}_p$, let $L/K$ be a finite abelian Galois extension of odd degree and let $O_L$ be the valuation ring of $L$. We define $A_{L/K}$ to be the unique fractional $O_L$-ideal with square equal to the inverse different of $L/K$. Combining a result of Erez with a result of Fainsilber and Morales we can see that $A_{L/K}$ admits an integral normal basis that is self-dual with respect to the trace form if and only if $L/K$ is at most weakly ramified. For $p$ an odd prime and $L/\mathbb{Q}_p$ contained in certain cyclotomic extensions, Erez has described such self-dual integral normal bases for $A_{L/\mathbb{Q}_p}$. Assuming $K/\Q_p$ to be unramified we generate odd abelian weakly ramified extensions of $K$ using Lubin-Tate formal groups. We then use Dwork's exponential power series to explicitly construct self-dual integral normal bases for the square-root of the inverse different in these extensions. These constructions generalise Erez's results for cyclotomic extensions.