Congrats on getting through a rough and grueling education!!!
I would sit and rent for another year or two. If you've just had another child, plus all the changes in your life that are about to come, why stress yourself out any more than you have to? Maybe renting a house might be nice, then you can slowly adjust to different living, and find out more about what you like and don't like in the features of a home. Think of it as a test drive to see how you like it. when i bought my place, i was not expecting all the little costs that add up. i had a lot to furnish, plus little things to buy all the time, as well as reno projects to do. It was an expensive time for me, and the process of house buying was stressful as well.
Take this time to make a future plan. What is important to you and your partner going down the road? Maybe you want to work in other areas/different locations which again would make buying a house a poor choice. What are some big goals or things that you would like to do over the next 5-10 years?
I would focus on getting your financial house in order. It sounds like in about 18 months you could knock out most, if not all of your student loans.
Also if you want to travel, why would you want to worry about what happens at your house? make your land lord worry about that for now.
Then i would focus on a down payment. shouldn't take too long at your incomes, but i don't know what sort of house values there are in your area to choose from