The allowance to dry oneself on Shabbos:
One who washes his face hands and feet in hot water which was heated from before Shabbos, or even [one] who bathes his entire body in the Tiberius springs, or in cold water, the [Sages] permitted him to dry himself with a towel or sheet and the like, even if one is particular during the week to squeeze these clothes from their absorbed liquid. Nevertheless it is best to dry oneself with an item which one is not particular to squeeze even during the week.
The reason that this does not involve a Rabbinical laundering prohibition: The Sages did not suspect that one may forget and come to squeeze the towel on Shabbos being that this decree would be a decree which is impossible to implement, as being that all bathers need to dry themselves, if they were to prohibit drying in essence they would be prohibiting people to bathe, which is a matter that is impossible to prevent people from doing. [This inability to withstand the decree is known from past experience as] one time the Sages decreed not to bathe [even in water heated by fire before Shabbos] due to the bath house attendees which would heat up the water on Shabbos while claiming that it was heated before Shabbos, and afterwards [they decreed against even bathing in the Tiberius springs being that people would bathe in fire heated water claiming that it was Tiberius spring water, and afterwards] the Sages saw that the public is unable to uphold the decree and they thus retracted and rescinded it.
Now, although today in which it is possible to abstain from bathing on Shabbos [and thus the decree is now one which can be kept by the public], and thus it is possible to decree against drying oneself, nevertheless drying oneself contains no prohibition being that it was never prohibited by the council of Sages [which are the only authority to make such decrees].
The allowance to carry the towel back home in an area with an Eiruv: It is permitted for the bather to hold the towel in his hand and bring it from the bathhouse to his home so long as there is an Eiruv [in his community] and the Sages did not suspect that he may forget and come to squeeze the towel during his return home (as if they were to suspect for this and would thus forbid against bringing the towel back home, this would in essence be prohibiting drying oneself on Shabbos , as it is not possible for one to disown his towel and leave it in the bathhouse)
In any event it is forbidden to give over the towel to the bath house attendees being that the attendees are suspected to squeeze [the towel] with their hands.
It is permitted to dry oneself with a cloth on Shabbos, although it is best to use a towel to do so rather than using clothing which one is particular to squeeze out their water upon them getting wet.
Should one dry himself with a towel on Shabbos rather than wear his clothing while still wet?
Yes. It is better to use a towel, as one’s clothing is not designated for drying with and he may therefore come to squeeze the water from it.