How do I improve email deliverability from a neglected domain?

It always starts with the list. You mention 'non opt-in' lists, which is certainly more troublesome than any domain name. Your best course of action is to get a new domain, re-confirm the lists you have (requiring the subscriber to click through in order to stay on), and only accept confirmed opt-in lists moving forward. It will decimate your mailing list in the short term, but long-term eliminate your deliverability concerns and improve ROI.

@Ben Isaacson. Thanks for you input on using a new domain.

I should have mentioned since last year that I require my clients to get permission from their customers to be on their list (and provide recommendations on how to ask for that permission).

On the page where they upload their .cvs list, there's an opening paragraph says, "We’re very careful not to include email addresses that have not given you permission to receive your newsletter. If you want to be certain of what can and can not be included, click here for examples." Then, there's a checkbox that says, "These contacts have given us permission to receive our newsletter."

I then have the list cleaned, upload into Interspire and a welcome autoresponder is sent out from my client. This welcome email tells the customer they're now on the monthly newsletter list, shares a link to previous newsletters and tells the customer they can unsubscribe anytime, including removal links in the header, the footer and a large one in the body that sits between the copy and my client's signature.

I just thought I should mention this. I didn't know better before, but experience and hard lessons make us smarter.

Cheers and thank you.

There are several methods that you can deploy which suits you the most, for example you can follow good email sending practices or maintaining a healthy email list or avoiding spam traps.
Besides if you do have any questions give me a call: https://clarity.fm/joy-brotonath