I think the engagement needs to be cut off and she needs to fix herself before you can ask her to marry you again. Because of the push-pull, you need to tell her you cannot marry her at this time and then go no contact for a minimum of a year. Tell her you will not be answering calls or seeing her during this time and she needs to use this time to get the help she needs. Perhaps in a year you may contact her again, maybe not. You will be open to a relationship with other people during this time also.

This is an unhealthy relationship and the current dynamic needs to stop. I suspect that without you supplying the attention she seems to demand she will move on to get that supplied somewhere else.