12v Supply for Uno, ethernet shield and relay

Let me explain how I’ve got things set up, and then you can tell me if there’s a better way of doing things.

Currently, I have an Arduino Uno with an ethernet shield attached. They are powered on the DC jack by a 12v “wall wort” (which in reality is probably putting out something like 13.5v with such a low load). I then have a relay, that requires 12v, connected to the Vin pin of the Arduino to power it.
It all works beautifully but the voltage regulator, unsurprisingly, is getting very hot.

What I’m looking for is a solution so I can power both the Arduino and sheild and the 12v relay from the same power source. Would it be acceptable to create a simple potential divider to drop the 12v down to, say, 7v before it goes to the Uno’s voltage regulator and have the same 12v supply running the relay?

Thanks in advance

You can not use a resistors divider to power the Arduino.

The best option is to use a DC-DC converter to lower the 12V to about 7.5V for the Arduino.
It is also possible to let the DC-DC converter make 5V and connect that to the 5V pin of the Arduino. That has a risk that the voltage regulator on the Arduino board will be blown.

You could replace the relays for 5V relay, so everything runs at 5V.

Could you use a regulated power supply of 12V ? A wall wart labeled "12V" could output maybe 18V without load.