Learn Spanish (Spain): Me encuentro mal.

You are right that encontrar normally means to find. However, when used as a reflexive verb with a pronoun like me (encontrarse), it can mean to feel (in terms of health) or to be located. Here, me encuentro translates to I feel.

Could I say Me siento mal instead?

Yes, absolutely! Me siento mal (from the verb sentirse) and me encuentro mal mean the exact same thing in this context. They are used interchangeably in Spain to say I feel ill or I feel bad.

Why is it mal and not malo?

Mal is an adverb meaning poorly or ill, and it describes how you feel (the action of the verb). Malo is an adjective meaning bad, which describes a noun (like a bad dog). Since you are describing your physical state and how you are doing, you use the adverb mal.