Your vehicle is not safe to drive.

Presumably you know this already but think it's ok because it's unlikely that you will end up in an accident as a result.

If you want to resolve the safety issue your lights are apparently now causing, you can have the entire headlamp assembly retrofit for LED bulbs, instead of just replacing the bulb. The reflector is designed for a particular type of bulb -- replacing just the bulb means there is often no way to aim the light in a safe and effective direction.

What kind of headlights do you have? If they're sealed-beam headlights, simply replacing them and aiming them will resolve the problem. if they're composite headlamps you almost certainly need to replace the lens and potentially the reflector if they don't give you any light at night.

These are sealed-beam. The problem is that they're so high off the ground that they are about eye level with most drivers, meaning they'll shine in their eyes more than other lights.

There are plenty of good sealed beam LEDs from reputable, American companies such as JW Speaker, Peterson, and Truck-Lite. Sample catalog below:

https://www.jwspeaker.com/products/categories/headlights/

I hope you chose something from one of the aforementioned companies. The aftermarket clones are just that...poor-quality clones.

Anyway, being able to see wet pavement and puddles is almost irrelevant. Wet road surfaces are good specular surfaces, which means that light from your headlights is bounced away from the car. Dry road surfaces, on the other hand, scatter light in every direction, and as a result, dry roads appear to be better lit than wet roads. However, simple physics dictates that the amount of light hitting a road surface doesn't change just because someone poured a few gallons of water onto the road. The amount of light hitting the road surface is a property intrinsic to the headlamp.

And really, there's way too much emphasis placed on how bright a wet road appears to a driver. Last time I checked, most obstacles (deer, pedestrians, disabled vehicles) were above the road even when the road is wet. The amount of light that is reflected off something above a wet road wouldn't change much whether the road is soaking wet or bone dry. And it's the light that is reflected off an obstacle that's important in perceiving those obstacles (well, that, and the contrast of the obstacle).