What does she want to do when she’s outside of her cage? You say she’s screaming to be let out- have you tried just letting her be out on a safe spot such as a playstand?
If a bird is screaming out of frustration- it’s a very easy fix if you can bring the frustration down. A bird screaming to be let out is either bored, or feels the need to be someplace other than the cage it’s currently in. The feelings can be so strong it leads to neurosis they can’t control.
First thing I would do is let her out. Where does she go? If she just wants to go to a window to look out at native birds? No big deal. Certainly allow her that, rather than talk of re-homing.
Many, many birds on here get their cages opened in the morning and are allowed to travel themselves or are loving placed on a tree/gym/playstand in front of a window (or some other interesting place in the household) and left there to enjoy their day. This is especially important if you have a singleton bird who innately feels lonely. They need to be able to be someplace other than their roost (the cage in which they sleep) or it can lead to neurotic repetitive behaviors, such as screaming. Birds do not naturally stay at their roost the majority of the day, they are hard wired to want to move. Add in needing to be around flockmates- whether human, furry, or other birds, and that drive is a powerful force.
I would just let her out, maybe on a day that’s low key and where people will be in and out of the home the majority of the day. See if she just wants to be somewhere other than caged. Hopefully you have a tree stand or play stand that has water, food, and toys available on it. If not, that would be a great investment.
I wouldn’t assume she isn’t safe to leave out- most Tiels actually are very good and don’t get into trouble, comparatively speaking.