Now that I know it's not natural to say “服を汚くした” given no results found, I am just wondering is it more common to directly use the verb itself than to change its adjective form into an adverb to modify する？
This is a very interesting question (your before-edit question included), and before I take a look at comparing your two specific examples, I'd like to take a look at first comparing 汚い and 汚れる(or rather, the present-state-describing 汚れている), which are in a sense the intransitive versions your examples derive from.
First, 汚れている. 汚れている in a literal, non-metaphorical sense is an objective statement/fact that something has an arbitrary amount of literal dirt/filth/grime on it. What is 汚れている to one person will generally be 汚れている to other people. This is why I prefer to translate it as "(something) has dirt/grime/filth on (it)".
On the other hand, 汚い is a subjective statement used to convey that the speaker feels that something has passed their personal comfort level of dirt/grime/filth. What is 汚い to one person may not be 汚い to another.
This often leads to 汚い implying a higher level of dirtiness than 汚れている, because any amount of dirt causes something to be 汚れている, but only after a certain amount will something be 汚い.
As an example, I often forget to clean the lenses on my glasses for extended periods of time. If someone were to say to me:
I might retort:
This is because while it may be true that my glasses have dust/oil/grime on them, I can still see, so to me they aren't 汚い.
However, if someone were to say to me and my same dirty glasses:
I might look them over and say:
This is because the fact that my glasses have dust/oil/grime on them is an objective fact; it cannot be refuted.
Going back to your two examples, 汚くした and 汚した, the same principal applies.
服を汚した implies that objectively, you got an arbitrary amount of dirt on your 服. However, 服を汚くした implies that the amount of dirt/whatever you got on your clothes has surpassed a certain threshold, and now you consider them to be beyond your comfort level of dirtiness.
The same thing applies to other adjective+く+する/verb pairs like 冷たくする/冷やす or 温かくする/温める. The adjective+く+する comes across as much more subjective than the plain verb.
Whether one is used over the other depends on whether you're overtly trying to be objective or subjective:
冷やした。 → I cooled (it) down. (It decreased in temperature a bit)
冷たくした。 → I cooled (it) down (to or past my threshold of what I consider 'cold').