Just keep riding and you'll be able to lengthen your time in the drops.
I think, because I started riding a road bike when I was 13 - decades ago - I don't have any problem staying down. I wouldn't say, at 171 pounds this morning, and 5 feet 11, that I'm overweight. On my bikes with the most drop, though, I can feel the top of my thighs touch my gut. It doesn't impede the time I can stay in the drops, though.
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