Does anything result from my performing tayammum and fasting, even though I was able to perform ablution, and I was not aware of the fatwa of Ibn 'Uthaymeen that whoever has water that is not sufficient for ablution should use it until it runs out, and if he does not complete the ablution, he should then perform tayammum?

Tayammum (dry ablution) is not valid during travel as long as water is available, because Allah Almighty says: ﴿…and you find no water, then perform Tayammum…﴾. You are required to make up the prayer if you performed Tayammum while water was available, or if the water you had was sufficient for all the organs of Wudu (ablution). Jurists differed on the ruling if the water is little and only sufficient for some of the organs: The Hanafis, Malikis, and most scholars held the view that one should only perform Tayammum, while the Shafi'is held the view that one should perform Wudu with the available water and then perform Tayammum. This second opinion is the more preferred one among the Hanbalis. However, the correct opinion is to suffice with Tayammum only, without combining it with Wudu in this situation.