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1: Your level of hormones changes "while" travelling.
2: Your level of hormones changes "during" travelling.

I think that "during" would be right if travelling is used as a gerund in the second sentence.

3

The conjunction while doesn't work with nouns and noun phrases. Thus, locutions such as while the house, while the pen and while your travels can't stand on their own and don't really make any sense. You need something that's active—something that's doing something. That's why with the conjunction while you can either use the "ing" form of a verb (present participles place special stress on the process of doing something) or a clause consisting of a subject and a verb (a verb is a doing word):

Your level of hormones changes while travelling.

He visited the Louvre while he was in Paris.

With the preposition during, on the other hand, you can only use a noun or a noun phrase (something that's stationary):

Your level of hormones changes during travel.

It's very quiet here during the night.

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