It is true that only washing “for enjoyment” is forbidden on Yom Kippur.[1] Therefore, one may use water to wash off dirt[2] or to carry out a required washing before davening,[3] and one also may get wet if he needs to pass through a body of water for an important purpose.[4]
Why the prohibition is so limited is a good question. After all, it clearly is not permitted to violate other aveirot simply because one does so for reasons other than enjoyment; one may not eat or drink on Yom Kippur for similar reasons, short of pikuach nefesh.[5] The simplest explanation is that the innuyim[6] of Yom Kippur, other than the prohibitions against eating and drinking (the two labeled as the major innuyim), are only Rabbinically mandated, and it is not uncommon for logical leniencies to be built into Rabbinical prohibitions. Indeed, several Rishonim, including Rabbeinu Tam,[7] cite such leniencies as proof that the minor innuyim are Rabbinic. The Rambam,[8] however, is among those who maintain that all the innuyim are of Torah origin. The Ran[9] explains that according to this view, we have here an example of Torah laws whose parameters were given over to the Rabbis to establish.
With that background, we will address your specific question. The Rosh[10] cites the following ruling of a Gaon: “If someone wants to wipe his face on Yom Kippur, if he is an istanis[11] and his mind is not at rest throughout the year until he wipes with water … he may wipe; but for others, it is forbidden.” In addition to apparently distinguishing between applying water for pleasure and fulfilling a specific need, this opinion considers the subjective frame of mind of the individual. The Shulchan Aruch[12] accepts this leniency. However, some Rishonim and Acharonim argue that there are problems with the application of the Gaon’s statement. The Maharil[13] cites but rejects the ruling, without mentioning why he does so. As often happens, the Rama[14] and Ashkenazi communities follow the Maharil. The Bach[15] posits that the Gaon and the Rosh (as well as the Tur) permitted wiping with water only in a case in which there is actual dirt on one’s face. (The Shulchan Aruch, however, apparently reasons that if the Gaon’s ruling referred to a case of actual dirt, there would be nothing noteworthy in his ruling. He must therefore mean that an istanis with a clean face is equivalent to a normal person with a dirty face.[16])
The Aruch HaShulchan[17] argues that the Rosh must be understood in context, which greatly lessens the leniency’s scope. The gemara[18] speaks about one who would dip a towel in water before Yom Kippur and then run that towel over his face on Yom Kippur, when it was less wet. (The Rama rejects using this system in our times out of concern that one might squeeze out water from the towel on Yom Kippur.[19]) The Aruch HaShulchan understood that the Gaon limited the gemara’s leniency, such that a regular person could use this system only to run the towel over his eyes, a most sensitive area that needs cleaning; only an istanis may use that system for the entire face. According to this view, the Gaon is not conveying a leniency, but actually somewhat of a stringency.
In any case, while Sephardim may be lenient as you suggested, Ashkenazim should not.[20] The practical logic for Ashkenazim seems to be that the refreshing feeling that wakes one up is considered washing of pleasure, even if the long-term interest of that pleasure is to help one concentrate on his davening. After all, being hungry is also often not conducive to kavana for davening, and a physically subdued feeling is not a contradiction to the Yom Kippur spirit. Only removal of negative extraneous materials from the body and incidental contact with water are included in the leniency of lack of enjoyment.
For you, we suggest considering putting your face next to an open freezer or placing something cold but dry on your face.[21] This may help.