Just my two cents, as someone who also struggles with making satisfactory-feeling precipitation maps:
Do you have a map with wind contours drawn on it for the updated version of the project? Precipitation decisions really depend on how you've decided the winds work out. For example, whether that desert in Artemis reaches the east coast depends on whether the east coast has a source of precipitation like being the receiver of a wind pattern originating in warm waters (and then depending on the direction of that wind the question is to what extent do factors prevent that water from reaching the desert portion, such as it being more tangential to the coastline and missing the desert region or the water all raining out before it gets to the desert). Mountain ranges don't generally stop deserts as far as I know; they're more likely to cause them (though they aren't the only possible cause) due to rain shadow effects causing water not to reach the desert side depending on wind directions. If there is any wind coming off warm waters reaching that east coast before other large land areas it should be seeing that precipitation. If the wind patterns from water bodies "miss" that area during a given season, or come overland through other areas that see precipitation from it before reaching that area, it should be drier. Given your desert positioning I'm assuming that region's centered on around 30N latitude; note that both the Sahara and Florida are around the same latitude but the latter gets wind off the Atlantic and varying pressure systems and the former has a high pressure system that stays mostly in place and gets most of its wind coming from the northeast (with mostly land in that direction). That's from some brief research but I'm not really knowledgeable about the complexities involved in those systems. As an overall summary, if the east coast of Artemis is getting wind coming from the ocean to the east it should be wet, like China getting wind off the Pacific; if its prevailing winds are coming from the west across the desert it should almost definitely be dry (and of course those statements apply for individual seasons if the wind patterns change over the course of the year).
When looking at rain shadows, it's not that the mountains are dry specifically; it's that on the rising slopes more water falls due to orographic lifting and then there isn't as much moisture left in the air by the time it reaches the other side. Mountains that exhibit strong rain-shadow effects probably have increased precipitation on one side and decreased on the other. Azélor's tutorial talks about orographic lifting increasing precipitation; just remember that anytime you increase the precipitation of an area due to elevation any area whose precipitation is fed by the same airstream further on has lost that extra precipitation that fell there. The more you increase the precipitation on the side water-bearing air is incident on, the more you should decrease it past that point.