I got some advice from another site that my climbing aloe has black spots from overwatering. This resulted in me repotting in better soil, applying a fungicide, and cutting off the areas with black spots. I also have not watered the plant in about a week.
Unfortunately, things have gotten worse: the leaves are now very curled, some are wrinkled, the pups are drying out, and the black spots are spreading. How do I heal this plant?

For reference, the soil is a mixture of general purpose succulent soil and inorganic media. It also sits in a West facing window but gets supplemented with grow lights.

It looks fine to me, just a little dehydrated. There is no disease to treat, that I can tell. I have never used fungicide on an aloe and I'd recommend you avoid removing leaves or leaf parts unless they let go on their own and are easy to grab. The curling of the leaves is pretty standard for a plant that's gone for a while without water. It should be reversible within a couple days after a proper deep watering if the roots are functioning well. If you're concerned about fungal diseases, the best prevention is achieved with the use of strong light (the maximum possible natural light, if this is an indoor plant), good air flow, and careful attention to watering.

Water well (until the soil is saturated and water comes out the hole at the bottom) and then wait for it to go dry at depth (not just at the surface) before watering again. Watering while the soil is still moist or wet increases the risk of rot, but there is no advantage to leaving the soil bone dry for any extended period.

Nice plant. Aloes can sometimes show this curling leave behavior after a repot. The plants can 'sulk' a bit whilst they settle in (often looking a bit worse before they improve). As long as you get the basics of care right (as advised above) the plant will usually bounce back with a little time and patience . Come back and let us know how you get on. Good luck
Steve

Thank you for the info!

I invested in a better grow light and moved the plant to a location with more light. I realized that this plant likes lots of light so hopefully I'm giving it what it needs.

It's been about a week since doing a thorough watering of the entire soil. Now the leaves are even more curled and have turned a darker shade of green. I assume this is probably a sign of underwatering (the only time I saw the leaves this dark green was when I first got the plant as it was severely underwatered) but I'm very hesitant to water again so soon. Do I just need to be more patient and wait for it to improve, or is this a sign that something else is wrong?

The green probably has nothing to do with watering. I would think that patience is the key right now.