Brady,
First off, welcome to the neighborhood. I hope you and yours will like living here.
I suppose my experience since moving here a couple years ago is similar to that of Jay, Glen and Robert in that I rarely have trouble with goatheads in the Wasatch bench/front/up-on-the-hill areas and mostly encounter them down in the valley especially through the corridor along Jordan River Trail and connecting trails like the Denver & Rio Grande Western Trail going north towards Ogden. However, I would note that most of Salt Lake City, its metro area and associated business/government/cultural services (e.g., breweries, bowling alleys, the Bicycle Collective, most but not all of the better Asian takeout food) are in the very same valley where all the goatheads lie. That is also where I live and do much of my riding and I get lots of goat head punctures.
Regarding armored tires, my partner's 'round-town bike has Schwalbe Marathon green guard (MGG) tires and hasn't had any flats, but it also hasn't been ridden a heck of a lot. The MGGs are heavy, stiff and basically just so damn thick along the tread that thorns can't reach all the way through to the tube. My trike came with Schwalbe Marathon Racers which have a nylon fabric belt under the tread. Thorns go through the belt pretty regularly, and the local thorns are long enough to get picked up in the sidewalls above the narrow belt. A Shikoro might fare better, but I recall various kevlar tires getting punctured by thorns in my distant youth in northern California as well.
My solution on the trike has been to use lightweight tires (Tioga Powerblock/Powerband S-Spec and Compass in ca. 40mm widths) with inner tubes containing Orange Seal---a solution cribbed from Patrick Moore's posts on this list and on iBOB (thank you Patrick!). With this setup I occasionally have a puncture which noticeably leaks sealant for several minutes and requires a few stops to add air as it leaks slowly before finally sealing. In the past year (9 mo. riding, about 4500 mi. on the odometer) there have been two or three times when I had a puncture that wouldn't seal enough while riding and I had to change tubes. Inspecting the tires and tubes at those times shows that dozens of punctures went unnoticed in that same period. Unfortunately that meant having to put a new tube into a tire the still had a score or so of thorns still sticking through it, which is about as much fun as it sounds. A few times I have gone out to the shed and found that a tire has gone flat overnight with all the sealant having leaked out between tube and tire through some hole making a big old mess.
I have not tried a tubeless setup yet. Maybe after my current tires wear out, though I don't know that any of my current rims are tubeless-compatible. My suspicion is that at least some of the poor sealing with tubes is due to the hole is the tube becoming misaligned with the hole in the tire, preventing the sealant from spitting out nicely and drying to form a plug. Also, I find that some of the seals or plugs in the tube can be pulled open upon removing the tube from the tire or even just on fully deflating a tire so that the tube pulls away from the casing. I imagine a tubeless setup could fix both of those problems.
I have observed that most of the thorns I pull out of my tires, at least away from the very center, are stuck in grooves/holes/corners of the tread pattern as though the thorn tip was sliding away until grabbed by the shape of the tread. Thus I suspect that slick tires may fare better w.r.t. thorns.
In short, goatheads can be a real scourge around here if you like to explore the low-lying (4500--5000 ft.) areas, but sealant in tubes can largely solve the problem. In addition, I think that using full-sized tubes (i.e., ones that need not stretch much to fill the tire) improves things further by helping to keep the tube and tire holes aligned, and that slick tread (e.g., as on your current tires) will help even more by avoiding some punctures in the first place.
Mark in Salt Lake City, Utah