Major = Happy, Minor = Sad, Can the opposite also be true? Options

[AK Rich]

Hey everyone. This is something that I wonder about from time to time.
Normally when we think of a happy or uplifting song, it tends to be a song in a Major key, right? And when we think of a sad or dark song, it tends to be a song in a Minor key, right?

My question to everyone is this. Can the opposite be true as well? Is it possible to have a sad sounding song written in a Major key, or a happy sounding song written in a Minor key? And if so can anyone provide any examples? Or describe some approaches that would enable this concept to work?

I am thinking that examples of this are rare but do exist but the only example I could come up with is a song in a Major key that sounds sad.
I think it may be less difficult to pull off making a Major key sound sad than it would be to make a Minor key sound happy. Or maybe I just haven't been able to wrap my mind around a way to make it work or think of a song where it has been accomplished.

For my example of a song in Major key that sounds sad, I submit to you a song from Frank Zappa, entitled "Watermelon in Easter Hay" from the album "Joe's Garage." As best I can tell, this song is in E Maj with only 2 chords which are A Maj and B Maj with an E for the bass note (B/E).
Or have I been fooled by some kind of music theory slight of hand?

https://www.youtube.com/watch?v=_3cu8sDa90Y

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