Thank you, Bass. That video had exactly the information I needed.
It's much more useful to be told what the limitations are for my set up than to be told I can make things better with a different set up.
For those who may find themselves in a similar place in the future, here are the details I needed from that video:
If you have a simple plug 'n play (PnP) microphone like me, you'll need to use WASAPI. ASIO may be better, but it isn't an option unless you have a separate audio board or a microphone with one built in.
The latency goes down if you use WASAPI exclusive, instead of WASAPI shared, but that does mean that nothing else on your computer can use your headphones or microphone while you are using it in Cakewalk. (You can select "Suspend Audio Engine When Cakewalk Is Not in Focus" so that you can tab away to say, a tutorial YouTube video.)
WASAPI can't automatically tell what latency adjustment to use like ASIO can, but there's a "Manual Offset" for each device where you can enter in a number to try to compensate for any remaining latency.
This last thing is really what I was looking for. In fact, I'd tried using this before, but I couldn't get it to work properly because the number here doesn't (at least on my system) seem to correspond directly to a millisecond delay, so the values I was setting were way too small to notice a difference. I had about a 90 ms delay (just enough to be noticeable and annoying), and I had to set this value to around 2000 to compensate. I don't know if this number is measuring in different units or there's something else going on, but it worked for me.
I expect I'll likely have to adjust this value periodically, but I'm okay with that annoyance.
