Not that it will help, but I use a vector application for my maps, and when placing rivers I use shaped stroke lines - meaning that the further end of a drawn line gets smaller to a point, so the river width diminishes as it gets closer to its source. Say I'm creating a river from the coast, I might use an 8 point line that diminishes to a single point on it's source end. Any rivers that I intend to merge into that larger river I use a smaller width, so 6 point and 4 point lines depict rivers merging near the center of the river length, and 2 point lines for smaller creeks anywhere along the river length and near the headwaters. Since you're using a paint program, I'd just choose a smaller pixel width line to draw in your smaller, joining river arcs.
Regarding the point mentioned about mountains and tectonic plates. Not all mountains are on coastal areas (though many are). Consider the Himalayan Mountains, though the Indian plate is smashing into the Asian plate, the Indian plate is actually connected by land to the Asian plate and causing the rise of the Himalayan mountains, those mountains are nowhere near the sea, more like south central region of Asia. Also look at the Caucasus Mountains, same thing. Look at the Alps, caused by Greece slamming into Europe and noting this region as earthquake prone. Even the Rockies in the US are far from the sea. The California coastal range on the other hand follows the close to the sea, tectonic plate caused mountain formation - so coastal ranges do exist, but cannot be considered the same for all mountain ranges.
With regard to the prevailing winds and rain shadows, look at the coastal range in Washington state and SW Canada. The west side of the mountains is considered a temperate rainforest, whereas the east side of the same range is arid and sometimes desert. Look at Kuaui, Hawaii. The north and eastern side of the island is rich in palm jungle, but the west and southern side of the island is bone dry - almost a desert.
If your mountains aren't that high in elevation, however, a rain shadow might not form, such is true only for fairly high elevation ranges. Something to consider.