Do you know what your jeep actually weighs fully loaded with gear and unloaded? If not, you will be guessing at tire pressures, Even the "chalk test" is just a guess because it is a subjective interpretation. It may be a very good guess, but a guess nonetheless.
With the weight of your jeep you can find your "ideal" street pressure using the Rim and Wheel Association Load Inflation Tables, which can be found here:
https://www.toyotires.com/media/pxcjubjs/application_of_load_inflation_tables_20200723.pdf
Another way to get to the same place is as follows:
2006 TJs came with either 30x9.50/15 B 96H tires or 245/75R16-E tires (Rubicons). The factory door stickers called for 29 psi for the 30" tires and 33 psi for the 245/75's, which have a diameter of 30.5" and are similar in size to your current tires but with a greater load rating.
You can use this calculator, which is based upon the data in the TRA Load Inflation Tables, to determine the tire pressure for your 31" tires that best corresponds to the factory recommendations.
https://tiresize.com/pressure-calculator/
When I use the calculator to compare the 245/75R16-E "Rubicon" tires with the 33 psi pressure recommended by Jeep with your 31x10.5R15-C tires, the TRA still recommends 33 psi.
When I use the same calculator to compare the 30x9.50R15-B "passenger" tires at 29 psi that most likely came on your jeep to your current 31x10.50R15-C tires, the TRA recommends 25 psi.
Start experimenting at 25 psi and go up from there as necessary depending on how much heavier your jeep is than the factory curb weight.
PS -
For sand pressure, try this neat trick from Harry Lewellyn, designer of Coyote deflators:
https://www.coyoteents.com/stuck-in-sand/
Tip: Do this in your driveway before you need to know.