Do words that act as nouns and adjectives in the same form constitute a particular part of speech class?

Solution 1:

Nominalized adjectives can be used as nouns. Two types of nominalization are found in English. One type requires the addition of a derivational suffix to create a noun. In the second case, English uses the same word as a noun without any additional morphology. This second process is referred to as zero-derivation1. An example of zero-derivation is the noun green in golf (referring to a putting-green) which is derived ultimately from the adjective green.

For examples, see this.

Solution 2:

A subset of this type of word would be nominals.

https://en.wikipedia.org/wiki/Nominal_(linguistics)

In linguistics, the term "nominalize" means "to convert into a noun." A common example uses the adjectives "good," "bad," and "ugly," which can become nouns in the phrase, "The good, the bad, and the ugly."

Lest I get called out, I'll add the caveat that this doesn't cover all of your conditions. I think it would be hard to find a term that explicitly denotes that the words can't be any other part of speech.