Praise be to Allah.
The scholars said that leftover water from which a dog has drunk and the saliva of dogs are naajis (unclean, impure), and that vessels and garments licked by a dog must be washed. It is reported in the Sunnah how a Muslim should purify these things if that happens.
Al-Bukhaari and Muslim reported from Abu Hurayrah (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) said: If a dog licks the vessel of any one of you, let him throw away its contents, then wash it seven times. Muslim added: the first time with earth.
(Al-Bukhaari bi Haashiyat al-Sindi, 1/44; Muslim, 1/234).
The meaning of the word translated here as lick means if a dog puts its tongue into water, or whatever, regardless of whether or not it actually drinks. The hadeeth refers to vessels, but the scholars did not distinguish between vessels and other things in this regard. Al-Iraaqi said: Vessels are mentioned because this is the usual scenario.
The vessel or garment must be washed seven times, one of them being with earth. This is the opinion of Ibn Abbaas and Abu Hurayrah (may Allaah be pleased with them), according to one report. It was also the opinion of Ibn Sirin, Tawus, al-Awza'i, al-Shafi'i, Ahmad, Thawr, and others.
(Al-Majmoo, 2/586; al-Mughni, 1/46 al-Muhallaa, 1/146; Nayl al-Awtaar, 1/74).
It is not the last washing that has to be done with earth, so there is no room for the objection mentioned in the question that this will make the clothes dirty. The clothes will be washed with water (after the washing with earth), and once they are dried and ironed, they will be cleaner than before. And Allaah knows best.