PLEASE keep her inside. I'm sure you know all the reasons. Many cats here who are allowed to roam during the day have gone missing and none have been found. None.
There are lots of coyotes coming around and I and others think these poor cats got killed and...well you know.
Raise the blinds during the day if you are home to keep the cat from bending them.
Talk to someone at a local shelter or adoption place (NOT a place that "sells" pets) for more advice.
I'm sure you'll get more good advice from the people here who have cats.
Our cats were always indoor cats and never missed being outside.
I don't know where you "heard" that shelters euthanize cats. If you must give her up, find a no-kill shelter. Please don't advertise in the paper, on line or on Craig's List. There are horror stories about what happens pets who are re-homed this way and it's not pretty.