There is no simple way of saying if the word “folded” is an adjective or a verb in these sentences. Linguists are still arguing about the concept of part of speech as applied to cases like this. So I'll try to give a reasonably theory-neutral description of how words like "folded" are used, and then summarize the mainstream part-of-speech analysis in the last section of my answer.
Terminology in this area is also not entirely fixed, so I'll refer to words like folded and eaten using the theory-neutral term "-ed/-en words".
Passive -ed/-en words can be interpreted in two major ways
There are clearly at least two fairly distinct ways to interpret these sentences.
Eventive/dynamic: The most likely interpretation of “The clothes were folded” in the first sentence is that it narrates an event: an action done by an unspecified actor. That is, it would be more or less equivalent to “Someone folded the clothes.”
Stative/static: But it could also describe a state resulting from an prior, already-finished action, more or less equivalent to “Someone had folded the clothes.”
Similarly, the second sentence is most likely to mean “He folded the clothes”, but it would be possible for it to mean something more like “He had folded the clothes”.
While these are, as I said, fairly distinct conceptually, they are often very similar in meaning, so in many cases it's not possible to determine which interpretation was intended by the speaker or writer of a sentence.
Factors that force a stative interpretation
There seem to be some things that cannot be combined with the eventive interpretation, and therefore force a stative interpretation. I know of two that might be relevant for "folded":
The adjectival negative prefix "un-": A sentence like "The clothes were unfolded" (meaning "the clothes were not yet folded"), has to be given a stative interpretation.
Using the word as a gradable adjective: there are various ways to do this. It includes using a comparative form, or putting the adverb "very" before the word. These are not very likely to be used with "folded", but if someone did say "The clothes were very folded" or "the clothes were more folded than I thought", only a stative interpretation would be possible.
Do these two interpretations correspond to different parts of speech?
In phrases describing a state, there is as far as I know consensus that the word “folded” should be classified as an adjective.
In phrases describing an event, the mainstream analysis is that “folded” should be classified as a verb. But other viewpoints do exist.
It’s my understanding that, while the term “participle” can be used to refer to both categories of -ed/-en words, it is often reserved for the words of this form that are eventive and classified as verb forms in mainstream grammar; the stative adjectives may be called something else like “participial adjectives” to mark them as a distinct type of word.