One of the first steps is that you might consider is talking in private to a social worker before you begin the separation. Do that to lessen the risk for things such as suicide by your partner or harm to yourself or your belongings. Do it with kindness in your heart and an open mind. It could be you will feel like it is as tough as walking on burning coals and that it might be a total waste of time. Both those things might be close to coming true but you will gain some insights to possible solutions despite that.
If this individual is truly incapable of working then the social worker can set into motion the steps needed to help them obtain disability income, medical coverage and housing. They can help place that individual into temporary shelter while the paperwork gets processed. Of course it is not a pleasant situation, such things are not comfortable for anyone involved in it. But there is a big difference between a situation being hopeless and one being very uncomfortable.
Social workers are found in many places, in city government, non-profits, churches, senior centers, medical and metal health clinics, shelters, food banks, YWCA, unemployment offices, through suicide hotline numbers, the outreach of police departments, etc. Make a connection and get advice on the possibilities.
Try to do it in a location where you can stay a little while and where the climate is not too harsh for the time of year. If need be for a transition period be willing to rent a small, short term, storage locker for them for 60 days or so to help secure their personal belongings until they get in a stable living situation. Be sure it is their name on the paperwork, not yours, you provide the cash but stay off the rental agreement. It makes a big difference to have personal possessions in a secured place if they do need to sleep in a shelter during the transitional phase.