I did a quick map search of some of the Seattle neighborhoods mentioned by others for rentals under $1600 per month and there are not currently a lot of choices. Rents are on the rise in many areas, and investors are buying up many well priced homes to take advantage of this.
If Seattle is your choice, it may be worth expanding your options and considering a purchase while rates and prices are still low.
Your husband may have enjoyed the U District as a younger single guy, but now with you and children possibly in your future some quieter areas may be worth considering. I live and work primarily on the eastside and commutes are not too bad. Buses run express routes downtown regularly and there are many good opportunities and great schools on this side of the lake.
One more thought, while Napa is well known for great wine, we have some fantastic wines here too.
I'm afraid that most close-in neighborhoods, north of the ship canal, will allow you a 1 bedroom, 700 SF apartment for $1,500. Homes will start at about $2,000 unless they are a mess. The rental market is extremely competitive in Seattle right now.
I have a flyer that I can send to you with rental links if you want to contact me through my website. I've lived and worked in Ravenna since .... way too long to express publicly.
Since you are from AZ you are probably very acclimated to the Sun. My husband loved WA so he had no issue with the Weather. I on the other hand had emotional and health issues caused by the damp dreary weather.
It's a good idea to rent first before buying, to make sure that the community and everything else fits with your lifestyle. Connect with a local agent to help you understand your options.
Have an amazing day!
If these are the reasons, you might want to look south of the city where things are more affordable. The rents here are going up, and you can find a place to rent/buy in someplace like West Seattle. It's a great "urban village" and just minutes to downtown. Depending on the time of day and where you live it could take less time to get downtown than the U district. You can live in an area that has the CA beach vibe, the small town treelined streets vibe, the semi rural (homesteads in the city) vibe or the fancy condo living vibe.
My office is near the U District, but I do a lot of business in West Seattle, because I am such a fan!