Shelters are typically super scary for cats and they are often judged unfairly. Each situation will be different but you should look around your shelter and ensure that there aren't factors contributing to the cat's stress. For example, cat housing that is too small, free roam rooms that are over capacity (or that have a bully cat in residence), housing cats near barking dogs or other loud noises, etc. If you can get a cat into a temporary foster home you might find that the cat is perfectly fine but that it was the shelter itself that was causing the problems. If you can get the cat into a home and he/she flourishes, have the foster take photos of the cat relaxing or videos of him/her playing so that people realize this cat is fine in a home.
Having a small sign on the cage saying that I'm stressed out, or super scared or whatever the case may be is good to let people know what's going on, but too much signage is a turn off. Providing a hide box to give the cat a place to get comfortable in, or providing a curtain covering part of the cage so he/she has some privacy can do wonders.
Dr. Brenda Griffin has a great webinar on stress reduction in cats that might be helpful to you. https://www.maddiesfund.org/stress-reduction-happy-and-healthy-shelter-cats.htm
Keep in mind that no matter what the issue is.... biting, not using the litter box consistently, etc. might actually not ever be a factor outside of a cage in a shelter. Giving that cat a more normal environment (whether an office or foster home or his own personal suite at the shelter) might help you figure those things out. If you're referring to outdoor cats, they shouldn't be in the shelter in the first place so I would recommend TNR back to his outdoor home, or if that's not currently allowed market them as a working cat to local businesses such as breweries, garden centers or nurseries, equestrian centers, etc.
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