I own a Trek District. I swapped out the original riser bars for some Nitto drop bars. Admittedly, I like the look of the drops better than the risers. The drop bars "feel" faster when I'm riding in the drops but I don't know if that's because they feel sporty or if I am actually faster with them. Most of my riding is around town. Anybody know how much of a speed advantage drops offer? I know I miss the added leverage of the risers for climbing hills. What are your thoughts? Anybody have any hard data?

Hard evidence, not really. Drop Bars have been the only choice for Road Racing for many decades. Slight variations of Drops ep. Randonneur, touring, dirt drop ... are also popular for long, more comfortable distance. They're shaped slightly spread-out.

I've the opposite experience than you. On one bike (more actually), I went to a sort(s) of "upright" and found that I'd lost some hand positions and feel more secure and used to drops. I'm not nudging you to stay with Drops or anything. You may want to give them a chance. My friend's done centurys on his Mountain Bike, flat bars.
NO question that drops provide a much more aero profile, it's not just your impression.

Better aerodynamics = faster for the same effort.

I don't know what you mean by having less leverage during climbs, though.

Many people who ride roadsters don't seem to understand how "drop" bars are supposed to function and ride around all the time with their hands on "the drops" as if they were welded there.

Watch pro road racers. Most of the time, the hands are on "the hoods"....The brake hoods. That's the "normal" position, and the one that should be set up for most of your riding.
The drops are used to sprint, get a lower position to cheat wind, and so on.
The normal position for climbing is with the hands on top of the bar, on the flat section. This lets you move back on the seat a bit for more leg leverage, and also results in an "open" chest for easier breathing.

that is good information. I've always wondered why road riders use the drop when going uphill. Silly riders.

Racers aren't the only ones with drop bars, too. Long-distance riders often use them, and often for the same reasons, even though they aren't going to be doing 35+ mph sprints. Not only is it nice to have the option of ducking out of the wind after eight hours of riding, but the variety of hand positions helps for relief, too.

Not all drop bars are the same, of course. Track bars, like a couple Nitto models, are really only good for riding in the drops. The tops aren't flat, but droop downward, which doesn't leave much of a platform for the hands and kink the wrist into uncomfortable angles. I've got a bike with that style of bar, and it's the only thing I hate about the whole bike. I do enjoy them while in the drops, though, partly because the curve of the tops helps leave more room for my arms.

Building on previous explantions:

The difference riding in the drops instead of on the hoods makes is about 2-3mph at maximum effort for a good rider. Even a big reduction in air resistance achieves only a slight speed up, because the power needed to overcome air resistance increases with the cube of speed. How fast your risers were depends on bar width and stem length - there's nothing magical about drops - you could set a flat bar bike up with a long stem and get similar aero, if you were willing to put up with the awkwardness of the thing.

Have the hoods always been the normal position? Somehow I'm under the impression that originally the drops were the "normal" position, and that riding in the hoods all the time is a fairly recent development. I can't immediately find any references to corroborate that belief though. It's possible I'm wrong.

The position of your handlebars relative to your saddle height is the most important consideration for comfort IMO and IME. People tend to overlook this.

Really low drop bars are uncomfortable to many people who would otherwise really like them! They only need to raise them a bit.

I've seen really low flat bars/risers that are probably very uncomfortable as well.

That said, I use a slightly curved back flat bar (almost like a Noah's arc bar) on my Surly LHT as opposed to the tradition drop bar you see on most of them.

Modern racing bikes tend to be more aggressive designs with the rider stretched out longer and lower. Today's hoods may well be where yesterdays drops were aerodynamically.

Back in the 70s or so, most casual road bikes (the ones with the stem-mounted shifters and the "suicide" brake levers) did not even have rubber "hoods" on the brake levers.
If you wanted to use them for a decent riding position, you had to improvise some sort of padding.

Higher-end bikes of that period came properly equipped.