Sleep deprivation is a real mental health crisis and nobody talks about it

We had our third kid 6 months ago and I'm running on fumes. Not the cute exhausted-dad-meme kind of fumes - the kind where I'm having intrusive thoughts, my mood is in the gutter, and I can't focus at work. My wife sleeps one night, I sleep the next. The randomness is killing me because I can't plan anything or recover. Went to my doc and she basically said "yeah that's rough, try to sleep when the baby sleeps." Cool advice, except a 3yo doesn't nap. Anyone else dealt with this? What actually helped?