Because of the delays for US passport processing. I wanted to see whether there is any way to either:
a) Apply for a new passport while keeping my current passport, with the current passport remaining active (so that I can continue to use it and travel on it while waiting for my new passport or
b) Apply for a "second passport" without sending in my current passport, with my current passport remaining valid.

The thing that I am trying to see is if there is a way to put my passport in the queue for being processed, while being able to travel internationally in the mean time.

I am still regretting making the mistake of waiting so close to my passport expiring to renew it. My passport was does not expire until next March. Unfortunately, the pandemic made this a really difficult time to have to deal with passport renewal.

I would have to say no on this.

It does not sound like you meet the criteria for a second passport. And to get a new passport when you already have one before, you either have to surrender that one or you have provide evidence of loss or stolen.

It (a) works at times at US embassies/consulates abroad. It’s not been unusual for me to hear ACS tell US citizens (or their parents) that State won’t cancel the applicant’s current, expiring US passport or their in-process-and-yet-to-be-issued renewal/additional passport in case the applicant needs to use their expiring US passport before the new passport arrives at station; and that the station is willing to return the expiring passport for use in the interim. In other words, there is some allowance in practice for a US passport renewal applicant to retain and use the expiring passport for most of the interim between application submission and renewal passport delivery to the applicant if there is a need for such. I saw this being done just a few months ago again at yet another one of our embassies.

If OP is renewing at a Passport Office, i.e. a State Department office, I believe that the State Department can allow this as well.

The problem is with COVID-19, many State Department services have been impacted, including ACS. Now, the best way to get a passport, in OP's case, is to renew by mail.

I am personally very aware of how our embassies/consulates have been impacted by this virus situation and include all such things in my assessment of how to go about things.

The OP’s passport doesn’t expire until around March 2021.

Even GPO is going to be ramping up production of passport books up to close to normal runs and gets on the bandwagon for State a couple days after the 4th of July. I suggest taking it as yet another sign that there will be more options to get consular services in person for passport issuance in the US and at US embassies/consulates by September/October 2020.

Actually, there are a couple of countries that I would travel to that would meet the criteria that would normally allow for a second passport. Strangely, they are among the few countries that still allow Americans to travel to them without major restrictions right now.

Then get the proof and apply for a second one. Your OP did not mention that explicitly.

what are those countries?

Some countries in West Asia, Central Asia, South Asia and South East Asia if having an Israeli stamp, but Israel doesn't as commonly stamp US passports pages as was the case years back.

You can get a 2yr validity limited second US passport if you prove a need to submit a passport for visa application for 1 trip while traveling on a second trip. I have done so myself. However, you must do so at a Passport Office, and those are currently only giving appointments on Monday & Thursdays to medical emergencies of immediate family members. I have even been trying to help a Doctor get an appointment for a passport renewal to travel to a patient to no avail.

Needing to procure multiple visas at about the same time is indeed often considered valid grounds for a secondary US passport.

They used to have mainly 2 years of validity, but now it’s routinely 4 years of validity for them.

The secondary passports can also be acquired via ACS at US embassies/consulates.

If you get a second passport, the validity is currently 4 years and you get this cryptic disclaimer on the last page of the passport.

I'm a bit confused about how renewals work, since 10 years and 4 years don't overlap cleanly.

Specifically, my second passport expires this year, while my primary passport expires next year.. Am I understanding correctly that when I renew my second passport this summer, it will get a new expiration date 4 years later (even though for 3 of those 4 years, my primary passport would have been expired), and then next year when I renew my primary passport, I'll get a new expiration date 10 years later?

Where are you applying? Inside or outside of the US? In-person or via mail?

I'll probably apply inside the US, at a passport office (since it's rare that I stay in the US for more than 2 weeks at a time, I probably won't have time to wait for a passport to get mailed back).