And the headlights only turn off once the high beams are activated.

Possibly issue with dimmer switch/multifunction.

Just to add, voltage to the headlamps is hot all the time. The headlamp switch is part of the ground circuit.

Ok where is the dimmer switch located, and can I do it myself?

That arm that you pull to brighten or dim is connected to dimmer switch/multifunction switch.
Disconnect the negative battery cable.
Disable the airbag.
Remove the steering wheel.
remove the steering column trim covers.
Remove the 2 torque head screws from the multifunction switch.
Remove the multifunction switch.

There is a safety issue because of the airbag. Safety is your responsibility. You will also probably need a steering wheel puller. You have to decide if this something you want to tackle?

The multifunction switch is number 2 in the picture.

Ok, it look like a lot of work, I might have a professional to look at that problem.

Heres another question: This car has winter tires in the front and the back, and I think it makes the car run funny, could that be the case, and if so what types of tires should I change to?

I dont know? The area I live in has very mild winters, we dont use winter tires.