You can take a line trimmer and chop the top off the seedlings, pretty easily. Burns calories and it is the least expensive thing you can do. You can even take a rake and rake the little seedlings spreading them out in the sun. I know Arizona a little bit and it is mighty tough for anything to become invasive.
Your instincts are good...don't use herbicides. Your water table down there is in big trouble and a little herbicide is a big deal for water quality.
Weeds are nothing more than a plant out of place. They do happen to be less demanding and can survive better than...ornamentals. If you can stay vigilant, go out once or twice a week to chop the tops off or using a 'hula hoe' disrupt the roots, that is all it will take to keep them in check.
The other thing I would do would be to find a facility that composts human sludge and sawdust. They should be testing 4 - 5 X and be able to give you a printout on test results. The final product should smell wonderful, like the earth. Be dark taupe in color, fine textured, no sticks, no rocks...this is all I use for my lawns and plant beds. This will turn your sandy soil into water holding, nutrient holding, organism rich soil. All you have to do is dump it on your soil and the organisms see a virtual neon sign saying "Best grub in AZ! For Free!"...they will come, eat this decomposed organic compost and take it back into your soil profile, mixing it for you.
Not only will this improve your AZ soils, hold moisture longer, hold nutrients longer, add nutrients and soil beneficial organisms...but it is the best stuff to smother seeds you don't want germinating. 2" on top of these seedlings will turn them into compost AND you'll be vastly improving your soil for the plants you want to prosper. Can't use it for vegetable gardening as it IS a little high in heavy metals. Of course 3/4 of our world uses human poo for fertilizer (rice paddies, etc.)
Doesn't matter what kind of soil one has, the ONLY way to improve any soil is by adding decomposed organic matter. That is a fact. The worst soil one can imagine can become totally productive with just the addition of decomposed organic matter. Maybe adding fertilizer...but fertilizer won't be used well by plants without the pH stabilizing, micro and macro-organisms that are there only because there is readily available food of decomposed organic matter.
Killing two birds with one stone: Smother your weeds, weed seed while feeding your soil. Let me know if you are able to find a supplier! My kids live down there and I could pass the info along.
Where in AZ are you? Please send pictures, we could help you with drainage issues as well which you should address soon. Are you buying this home? Send a base plan of your home and property with property lines, utilities, outbuildings, neighbors, fences, drive way, walkways, vehicle access, North orientation, plants that exist and please include at least 3 - 6 elevations. This means the height at 6 different spots in relation to...the elevation of your home. You can get a transit, laser to do this. Ask a friend in construction to do this for you in exchange for washing their car, making dinner and washing dishes.
Look through your mortgage documents for an 'As-Built'...most of this should already be accomplished. Find out if you're foundation is 'slab', 'pier', foundation with crawl space or basement. What did they do for drainage when they built your home? Foundation drainage, asphalt emulsion, anything you can find about what you own and how it was constructed. Take pictures of your basement, foundation, your property with high and low points. Use graph paper, 1/4" to one foot or whatever scale makes sense to you. Take a picture of your drawing and send it. Let us see what we can do with that information.