Couple of thoughts.

A question about your inseam - is it your pants inseam or "cycling inseam" (look it up). The reason I ask is that if it's 31, that would imply, to me, that your legs are a little shorter than average, making your upper body a little longer than average. I'm ~ 5-8, have a 32" cycling inseam (30" pants) and ride what is typical for a 54cm frame - horizontal top tube measurement of about 54-54.5 cm. I use a 100-110 mm stem. I have my bars about level with my saddle, but used to ride them a bit lower (I'm 66). I believe as you get used to the bike you'll want a stem a lot longer than 75mm and a bit lower, depending on your comfort and flexibility.

But that's neither here nor there because you should adjust the bike in whatever way makes you comfortable.

The best way to figure this out is to have a real bike fitting. Not just a sales person talking to you during the purchase / test ride process and swapping out the stem without addressing all fit adjustments (starting with saddle position - see below). Ask at the shop if they do real bike fitting and have them do it. Or find a shop that does. Bike fitters have stems that adjust both height and length and really help you get that in the ball park (but may change over time....again see below).

That said, you can also do a lot of trial and error on your own and reading about fit online. Stem length and height is done after you get the proper position of the saddle. You adjust saddle position vis a vis pedals, both vertically and horizontally first. You also need to get the angle of the saddle correct for you - usually dead level or a tiny bit nose up. That gets your balanced position on the bike and then can really assess how you feel for the length and angle of the stem.

Stems are cheap on eBay. Not super light weight expensive ones, but plenty good, slightly heavier inexpensive ones. Think of a rough idea of what you think you want for length and upward angle and buy some and try them out. Resell after you've figured out what you need and your net cost won't be bad. Or buy a couple lengths of adjustable stem which have upward angle adjustments.

But I'd suggest the fit and also some patience realizing that if you keep riding, I think you'll find the length and height of your stem will change as you get used to the comfortable fit of a road bike.

Interestingly, the past few years I had some pretty bad back problems. Sciatica caused by bulging disks and other physical defects of my lower spine. i could not walk more than 1/4 mile at the worst. But I could ride my road bike and actually found that increasing the length of my stems about 10mm which stretched me out a little, felt better - my back wasn't as hunched as much and it just worked better. Just fwiw. Now I can walk/hike as far as my fitness allows with no stops for stretching, etc.

Keep at it and enjoy the new bike.