Is holding verses of the Qur'an with a hand that is likely to be contaminated considered belittling and disrespecting the Qur'an, and does this act lead to disbelief?

What happened to you is not considered disbelief or irreverence; rather, your motive was to respect and protect the Quran. The mere presence of moisture on the tissue does not signify its impurity unless it was in an impure place, and the default state of things is purity. Therefore, ward off obsessive thoughts.