Your environmental conditions will preclude finding a machine suitable for your purposes in the budget specified.
Humidity is a problem with many material types, especially nylon, but also with PLA and ABS, the more common filaments used in 3D printing.
You can likely reject PLA for your mechanical needs, as it is brittle and weak compared to ABS. PLA releases virtually no gases of concern, while some find ABS fumes to be offensive and dangerous.
The humidity issue is forefront in your search. You may have to construct within the garage a chamber in which you would operate a portable or window air conditioner unit, to keep the humidity in check. If you can assign a different budget to such a construction, that will leave your printer funding intact and better able to address your goal.
Selective Laser Sintering using nylon powder, also susceptible to humidity, which is sintered by a laser, hence the name, making very detailed and strong parts. The process is also self-supporting, allowing for fairly intricate parts. Once the machine is calibrated, the part accuracy can be quite good. Unfortunately, SLS machines are also out of the budget range you've noted.
You can use an external service to print the parts you design, at least at first, to get a better indication of how the various materials will work for you. Start with PLA, then move to ABS for a set of test parts, and even perhaps have some printed using SLS.
If you find, for example, that ABS will be strong enough, you might find an affordable 3D printer which will generate parts on your budget and timeline. For printing ABS, the warmer temperatures are to your advantage, but the humidity has to be properly addressed in any home/shop/garage installation.