We miraculously managed to get a good deal that we could blow all our savings on to get this new HVAC unit installed about a year ago. In the last few days, it's suddenly started producing a sour smell, particularly in my room. Looking up the issue, all I see is people saying to call someone to get it cleaned because it's mold and/or bacteria build-up. I changed the filter, but the one that was there was barely dirty at all, so I don't think that was much of a fix.
I can't go into detail, but there's a lot going on right now and we can't afford much. We barely even have time to organize fixing something like this. But I don't want to take risks with our health.
Can I get some advice please? What would y'all do?
*also my god the youtube algorithm-fueled thumbnails that come up when searching for tutorials on what to do about this are truly horrendous, look at this shit, just straight up embarrassing

What kind of system is it? Is it a central system where there is one coil with ducts going to each room or is it a split system with a seperate AC in each room?
As far as cleaning it goes, normally I advise only having it professionally done because the chemicals used can damage things if used incorrectly but when it comes to the indoor coil that is less of an issue as the chemicals involved in cleaning that are much more mild. The biggest issue you'll have is getting access to the coil itself depending on what kind of system you have.
A cleaning shouldn't be horribly expensive to have done professionally if you want to go that route. I'm a commercial refrigeration mechanic so my prices are much higher than most residential HVAC contractors. A coil cleaning and system checkup on a regular central air style system would typically cost one of my customers about $300. But like I said, that's commercial pricing. A residential contractor should be cheaper. I would recommend having it done professionally because while they're in there the service tech should also be doing a quick check of other components to catch any major problems before they happen.
But if that isn't in the budget right now and you do want to do it yourself, basically the process to clean an indoor coil is just -
Shut down the unit. If this is a central system then that means shutting off your airhandler or furnace. On a split system it means shutting off power to that indoor unit usually via the breaker panel. Use a non contact voltage detector (like $10 at any hardware store) to verify that the system has no power going to it before proceding.
Get access to the coil. If you're lucky then this just means taking off an access panel on your air handler, if you're not then this mean pulling apart a section of duct or disassembling a split system head unit. This is usually the hardest step. Pay attention to how you're pulling everything apart to ensure you can put it back togeather again.
Remove as much physical debris from the coil by hand as possible. Do not bend the fins or use any metal tools. You can use a coil cleaning brush or other long soft bristled brush to brush out the coil if need. Most of the debris however will becin the condensate pan under the coil and can just be removed with a shopvac.
Spray down the coil with a coil cleaning spray. I frequently use Nu-Calgon Evap Foam No Rinse (lookup part number 4171-75). It comes in an can and you just shake it up and spray down the coil. The goal is to use the pressure of the can to spray in between the fins of the coil where it will foam up and push debris out of the coil. This stuff will also kill anything growing on the coil. Once applied you don't need to rinse it off because it's designed to be rinsed off just by the condensation that builds up on the coil when the system runs. This cleaner shouldn't hurt your skin or any surfaces but I highly recommend eye protection because it can spatter a bit while spraying it.
Put everything back together and run the system. If the smell was coming from the coil then it should begin to get better. It may take a day for the condensate to fully rinse the cleaner off the coil though so give it time.
As far as preventing this problem in the future the best thing to do is to keep up on your filter changes and ensure that you are using the correct filters. Another thing that can be done is you can have a UVC bulb installed in the duct by the coil to regularly sterilize it and prevent anything from growing on it. However that really really should only be done by a professional because UVC lights are no joke and can cause severe damage to eyes, skin, and surfaces.