During the rinse and spin cycle, water is sprayed everwhere

It is normal for a washer to spin and spray fresh water over the clothes during rinse period. I don’t know why it would leave clothes wet. Look at your options. See if you can turn off rinse n spin to just spin.

Could be a failed lid switch. This is a safety switch that shuts the machine off during cycles.

I would look at lid switch on this model and plunger to lid .Under lid is spray flume that can be damaged by large loads ,check that close. put it in wash full and drop lid hard it may come on.

What is the model number of your unit?

The timer will always advance to the next program if the machine does not fully drain. There is ALWAYS a spin when draining the water out, there is only one pump and one motor. The waterlogged laundry maybe telling the timer that there is still water in the tub. Put the machine on "Spin" and cheat the lid switch into thinking that the lid is closed. Use a clothes peg or wide ended screwdriver and see if there really is a spin....

Yes, a magnetic lid switch and it is defective.
Replace it.

If you are getting no water spray during the rinse cycle that would cause the soapy clothes issue.

Are you getting cold water during the wash/rinse cycles? If you have the machine set to warm/warm you may only be getting hot water while the final spin is calling for cold.

If you have cold water available then the timer is not calling for the water spray. The problem is either in the mixing valve or the timer.

most new washers have cold water only rinse and spray rinse ck cold water coming into machine during wash if no cold comes in suspect cold side of fill vale not opening ..could be wiring issue or a bad valve