Congrats on your new beardie!
You mentioned being worried that he is not eating right now. This is okay, they can go through a bit of relocation stress. Once you get your temps and lights all figured out and your little guy gets more adjusted to his new home he will start eating for you.
I would like to suggest a few things, if that's okay!
Your tank looks very nice and that log is awesome, but I'm afraid from the pic, that your beardie may be too close to the basking light.
How close is he to the basking light? What watt is the basking bulb?
Is that a 10 gallon tank? It could be larger, but looks to be small in the pic. If it is a 10 gal, they can be very hard to get the temp gradients that you need, due to their size.
Glad to hear you are going to get a digital thermometer. You will definitely need one for accurate temps.
As far as your temps go, for a baby, the basking side should be between 100-110. A little more towards the higher end of those temps for a baby beardie. The coolside between 78-82.
What brand tube uvb are you using?