@DKW2DO sounds like you are in a tough situation, but that you still have a little bit of hope left that you can find a solution. It is great that you have the relisience to keep trying to resolve this despite all that is going on for you.
Depending on which state you live in, you may be able to apply for a housing swap, or transfer. Transfers need valid reasons - but might be quicker time frame than a straight swap. These are some example reasons that they may consider https://www.facs.nsw.gov.au/housing/living/move-swap-transfer/tenant-transfers If you do need to move out of your public housing property it may be that you can apply for a transfer before moving into temporary accommodation & so may be higher up the waitlist than if you had to reapply down the track.
As others have mentioned - it may help your situation with your public housing if you can get evidence from a GP or psychologist or whatever mental health professionals know about your situation to back up your need to move. Many public housing authorities have a specific form they'd prefer you to use for reporting these medical needs.
Also from what you have said I wouldn't rule out the possibility of negotiation to rent a room with your friend. It sounds like her sister may have concerns, and want to protect your friend from potential problems occuring that may upset their mental health. Your friend may have some of her own triggers that her sister is worried about, but a discussion beforehand could allow you all to work through any concerns to come to a mutually acceptable solution. Could you have a sit-down discussion with your friend and her sister to discuss further and you could put in place some house rules to make the expectations clearer of how it would work to live together? Possibly a trial period could be considered before you give up your Public housing tenancy?
Do you have any options to briefly stay in other accommodation for a night or few to give you a break from being triggered? Family, friends, couch-surfing, hostel or anywhere that you would feel safer than your current place, but that is affordable for you? May not be a great long-term option, but may give you a little relief from the triggering situation while you work things out.
Good luck - hope you can find a solution.