How do I clean up a moldy mess?

Anything upholstered is a loss. But wood can be cleaned up. Use bleach water to wash the furniture. Then just for safety sake, wash it with vinegar and water as well. Place it in the sunlight to dry out. Wear a mask if it's really bad.

I agree with the above response. Just make sure you get all the bleach off before putting it in the sun ( your wood may crack). Then after it’s dry use a good furniture polish, your wood will thank you 😊

You can try putting it outside on nice days. Cover it in baking soda, let sit as long as possible, vacuum, repeat. Small bowls of vinegar in the enclosed areas such as drawers and doors help too. It takes a few days to get the stink out, but worth the effort if you really like the furniture. I also use lavender essential oils on a cloth to wipe out drawers,interior chubby spaces, below, and behind after the baking soda and vinegar treatment.

Charcoal in the drawers is great for removing odor. I purchased these bags but I've heard you can put down newspaper and charcoal briquettes as well:

https://www.amazon.com/gp/product/B01N7NHVZK/ref=oh_aui_detailpage_o01_s00?ie=UTF8&psc=1

Check out my blog post: https://karupp-did.net/sorry-for-the-stink/

I had a horrible time with a dresser I refinished. This wasn't just musty...it was also cat which is ridiculous to get out of furniture. Check it out...I would say if it is musty...go for it. CAT....maybe think again. But check out the blog post because I go into exactly what I did and Yes, I eventually was able to get the cat smell out.

Vinegar solves 99% of your problems. Vinegar kills mold spores.

https://restorationmasterfinder.com/restoration/what-kills-mold-bleach-vs-vinegar/

https://www.servicemasterrestore.com/blog/mold-damage/mold-myths-will-vinegar-kill-mold/

https://www.servicemaster-restoration-cleaning.com/2017/03/30/mold-myths/

Wipe all the furniture down with a vinegar solution, rinse dry, and if possible, sit in the sun for a day or two.

If the upholstered pieces are bad, if they have good bones, you can rip them apart and reupholster them. If you are not up to the task, instead of junking and filling up a landfill, offer to DIY'ERS to work with. They will love you.

Coffee grounds, baking soda, newspaper, activated charcoal also help.