The Shulchan Aruch[2] deals with the issue of one who leaves the bathroom and is ready to eat. Such a person has two obligations of netilat yadayim – one as a result of using the bathroom and one in preparation for his bread meal. If he does netilat yadayim only once for both needs, he will necessarily create a problem of hefsek.[3] Whether he washes and then recites the beracha of Asher Yatzar for attending to his needs or first recites the beracha on the netilat yadayim, followed by HaMotzi, and only afterward recites Asher Yatzar, there will be a break between the time the other beracha became necessary and when it was recited.[4] Therefore, the Shulchan Aruch says to wash two times in such a case, the first to be followed by Asher Yatzar and the second by “… al netilat yadayim.” We can ostensibly learn from this ruling that the fact that one just washed his hands does not preclude his performing another formal washing along with its beracha before eating bread.
In reality, however, there is a machloket regarding whether netilat yadayim, when performed before a bread meal, is a mitzva that warrants a beracha when it is a repeat of a netila that was done recently. The Beit Yosef[5] infers from Tosafot’s discussion[6] regarding the two washings that we do on Seder night that if one does netilat yadayim for a low-level obligation (e.g., before eating a food other than bread dipped in certain liquids) and then needs to do one for bread, the latter netilat yadayim is still a full obligation that warrants a beracha. However, the Darchei Moshe[7] maintains that this is so only when a significant amount of time has passed, as we assume that the person eventually lost concentration in maintaining the cleanliness of his hands. If cases in which the washings were in close proximity, the Darchei Moshe is not convinced that netilat yadayim is required. Therefore, while one should wash a second time, he should not make a beracha on the second netilat yadayim.[8]
However, in our case, it is relatively simple to ensure that the “ritual washing” is adding something that the washing with soap and water is missing – that is, if only the second washing satisfies the halachic requirement of using a cup. We see this distinction in the halacha cited above regarding one who washes after using the bathroom and before eating bread. The Mishna Berura[9] cites Acharonim who say that the first washing, done to enable reciting Asher Yatzar, should be performed in a manner that suffices only for the purpose of cleanliness but is not halachically valid for a bread meal. If one did a complete netilat yadayim the first time, these Acharonim maintain that the second washing would need to be done without a beracha. An incomplete netilat yadayim can be accomplished by not using a valid washing cup, which is a requirement for netilat yadayim.
One could also make a second netilat yadayim necessary by having his hands touch a covered part of his body or some other “dirty” place.[10] However, we imagine that this second approach would not sit well with those who are particularly concerned about hygiene.
In addition to not using a cup during the hygienic washing, which is easy enough, one should preferably not use the first spurt of water from the faucet, as some authorities feel that this is equivalent to washing with a standard utensil.[11]
Turning to your second question: should one dry his hands in between hygienic cleaning and netilat yadayim? The reason that doing so might be necessary is that the water on the hands may be tamei,[12] which could prevent the water of netilat yadayim from purifying the hands. A similar case is discussed by the poskim. The Shulchan Aruch[13] discusses pouring some of the netilat yadayim water on the hands to remove dirt prior to the main netila. The Bi’ur Halacha[14] notes that there is no mention of a need to dry the hands before proceeding. He says that this is so because either the prewashing is part of the netilat yadayim process, whereby water can become tamei and is removed by the second rinse, or, to the contrary, the water that gets on the hands is not tamei unless it got there through an improperly executed netila.[15] While the Yalkut Yosef[16] agrees with this approach, there are significant poskim, both Ashkenazi and Sephardi, who disagree.[17] They maintain that if water is used to remove dirt prior to the netila, one should dry his hands from the questionable water before commencing the real netilat yadayim. Without getting into the intricacies, it seems that our case is more lenient than the one these poskim discuss. It thus appears that it is not necessary to dry hands that were washed for hygienic reasons before netilat yadayim.