Ignition and clock issues - HELP? PLEASE?

Hello! long time listener, first time caller.
I drive a 2005 Prius I bought second (or third or fourth) hand from a very sketchy guy a couple years ago. I have 187,800 miles (exactly!). My coolant control valve is a hot mess and has been since I bought this, and my hybrid battery leaves something to be desired, but for the most part I get good mileage and I get where I need to go. But I have some new issues and no money and I’m interested in some community help until I can afford to get a newer car.

In the last 2-3 weeks, my car sometimes fails to turn off when I park. I usually just hit the power button, but whether I put it in park or not before I press the power button, it will turn off and then cycle right back on and I have to power off again.

Around the same time but not exactly, the dashboard clock has been weird. I turned on my car after work and the clock was about 3 hours early. “That’s weird,” I thought, and reset it - I usually keep it late and my roommate had just commented that the time varied from 2-10 minutes late. I figured it was because I had just replaced the coolant control valve and resetting the battery had done something weird, but now every time I turn the car on it’s a different weird time. Last night it was about 75 minutes early, then this morning it was about 7 HOURS early, and now it’s just about an hour off.

And finally, most alarmingly, my car refused to turn on for a couple minutes. It was off for a few hours, then I drove to the store, and when I got back in, it unlocked fine (so, not the battery) but wouldn’t turn on. I tried the power button with my foot on and off the break, and with the key in and out of the ignition. When I was texting a friend for a jump, the alarm went off. I put my foot on the brake and hit the button, and it turned on partially (all the dashboard lights on, gas gauge empty), and a second press of the button brought it back to life. I am currently afraid to turn it off.

Any help you all have would be GREATLY appreciated as I can’t really afford a mechanic unless I know exactly what I’m looking for going in.

Thank you!!

 

Do you think that might also be the clock and the failure to turn on?

 

Update: I drive my car probably 15-20 hours a week. The 12V battery is about a year and a half old but I just had it checked at a mechanic and they said it’s ok.
I got a new error code today, the big red triangle/exclamation point on the dash, and a car with an exclamation point on the multi whatever screen. the manual says it’s a hybrid system error.