Is striving to call the Adhan in the mosque and lead people in prayer while traveling considered a form of showing off for someone whom Allah has blessed with a beautiful voice, and how can one avoid showing off?

A beautiful voice is a blessing to be used in seeking Allah's pleasure. Ostentation (riya') can be in performing an act or in refraining from one, and it is not correct to abandon a righteous deed for fear of ostentation. The call to prayer (adhan) and leading people in prayer (imamah) are not ostentation unless the intention behind them is to seek people's praise. To avoid ostentation, be conscious of Allah's watchful eye and strive against your soul to achieve sincerity.