Is it permissible to search for another wife without informing the current fiancée, given the discomfort with her, and acknowledging the urgent desire to marry and the weakness of faith?

The questioner must repent from sins and abandon whatever leads to them. He must also be wary of dying while committing a sin.

If merely touching a non-mahram woman is a grave matter, then what about that which is greater?

Marriage is obligatory for anyone who fears falling into indecency.

As for breaking off an engagement, the general rule is that it is permissible, though it is undesirable without a reason due to the harm it entails. If the girl is religious and of good character, one can repeat the Istikhara prayer and consult people of goodness. If she is not, then leaving her is preferable.

It is permissible to seek another girl for marriage through legitimate means during the engagement period.