How can I get sponsors or paid sponsors for my newsletter?

I recommend promoting the YouTube channel via the newsletter and the newsletter via the YouTube channel. With your niche being highly targeted this tactic should improve your reach. I also recommend starting a contest and offering gifts those that can generate the most sponsors. I'm sure your subscribers know of people that would sponsor your newsletter.

2,500 does sound quite low for an email list. That's not a criticism, just to say keep doing what you are doing to build up the numbers in the meantime.

In terms of getting sponsors I think this is a case of good old fashioned sales prospecting. Spend the time building up a list of potential sponsors that you would love to have on board. Create a spreadsheet or add them into your CRM with as much info as you can about each company. Do you know the contact you should be speaking to at the company? Can you get their email address? LinkedIn can be great for this.

Build a defined product offer so when you start speaking to these people you have a clear set of benefits to talk about and a defined pricing structure.

From there create an email series to send out to these prospects but be aware that you will NEED to follow this up with telesales activity to close the deal. Don't expect success on the first call but a consistent approach over a few weeks should start to wield results. Good luck.

You said "we use some affiliate links in the newsletter but they don't really bring in any revenue"
If your affiliate links aren't bringing in revenue what makes you think and ad will bring revenue to and advertiser?

You said you have a list of 2500 targeted people, what is the open rate for the newsletter and the click through rate, how many click on the links.

By Newsletter what do you mean? Is it an old school newsletter filled with a bunch of stuff. If so that could be part of the problem with the affiliates. Newsletters can be overwhelming and too much stuff for our information overload society.

Try sending one topic per email at one per week. 80% value & 20% promotion, meaning either 8 out of 10 emails provide value and only two promote something your's or someone else's, or 80% of each email is value and only 20% promotes something.

If your subscribers on YT are active & you videos have lots of views to the end, that may be a better place to look sponsors. Tim Schmoyer of Video Creators has some videos on how to do that. Also I would use the channel to continue to build the email list.

Guard you list and only present them things you are willing to back up 100%, if you guard them they will become your raving fans and will buy & promote for you. Continually put things in front of them to buy especially from other people and they will grow weary and move on.

Schedule a time if you'd like help deciphering your analytics or reviewing your content.

I have a lot of experience with email marketing. More than I wish I had sometimes!

2,500 is on the small side, yes, but it doesn't mean there isn't value there. As an advertiser, I've had success with smaller lists, it all depends on your historical performance rates, and what the offer is.

Is your list growing? What are your average open rates?

It's probably too small of a list (now) to start effectively selling to advertisers, but you might want to try some ad networks who specialize in email marketing.

Here are two:

https://liveintent.com/
https://www.syndicateads.net/ - they do email sponsorships, as well as other types. This company is owned by https://www.buysellads.com/. They would sell sponsorships for your newsletter on your behalf.

To grow your list I recommend using one of the following solutions:
http://optinmonster.com/
https://sumome.com/

Hope this helps!