I hope the job works out for you. Sounds like you really need a break.

It may be a moot point now, but you might want to understand a bit about how jail works. (I worked in one.) The police don't run the jail. They just bring people there. They can arrest you for breaking the law and drop you off at the jail, pending your appearance before a judge, who will decide what to do with you. Once the cops drop you off, they are completely out of the picture, except, maybe, to show up at court to testify against you. Sometimes cops do bring people to jail mainly for their own protection. If you were stumbling around the streets drunk or sleeping on a city bench intoxicated. The cops might arrest you for public intoxication. The penalty for that is often one night in jail. They arrest you and bring you to jail. Next morning, you go to court and the judge deems that you have already served your one-night sentence and tells you that you are free to get your stuff and be released. You can be arrested for other things, like being a public nuisance, where the cops main concern might actually be your welfare.

At the jail, you would get looked over by a paramedic or a nurse. They would ask about your physical and mental health. If you told them you were very depressed and in danger of self-harming, they would place you under special observation.

At any rate, you have to have violated in some way for the police to arrest you and bring you to jail. Once the jail accepts custody of you (which they can refuse to do, if they think you need emergency room care) the cops leave and have nothing to do with what happens after that.

As you seem to know, cops can just decide to bring you to a hospital emergency room, or to a psych emergency facility.

Some jails (larger ones) have psych units for inmates with unstabilized psych problems.