Thanks for the expanded map. I have a few more things for you to think about.

Why is the rest of the map unexplored? What's stopped people? If they're anything like the High or Late Medieval period, they'll have seafaring ships. A large enough desert may prevent expansion on foot, as will hostile civilisations, and perhaps storms on the east shore stop people from sailing down too far, but what stops people from sailing west and around the desert?

Monsoon climates really only happen on large enough landmasses where the pressure system is more or less fully affected by the sheer size of the land. Asia is big enough and at the right altitude to completely change its pressure system from a strong high to a strong low between summer and winter, and though North America is in the right position, it's too small for monsoons.

By moving the equator, I suggest taking the layer it's attached to (assuming it has its own layer) and stretching it downward until the equator is about 30 degrees south of where it currently is. First, though, correct your labels. The Inter-Tropical Convergence Zone (ITCZ) IS the equator, and the Sub-Tropical High Pressure Zones (STHZ) ARE the tropics of cancer and capricorn. You have them labelled as the space between. This will only confuse you. The ITCZ is always tropical and the STHZ is always desert on mid-western shores. South America isn't a desert, you'll note, even though it's in an STHZ zone, because it's too thin and the Andes changes the climate because it's such a huge range of mountains.

I suggest trying out these changes and then give the Climate Cookbook another shot - jc.tech-galaxy.com/bricka/climate_cookbook.html - and please let me know if you disagree or if I didn't make sense. Good luck