Dallas, Austin, Atlanta and Phoenix are all excellent, growing markets, so great choice picking those. I would also look into Charlotte, Raleigh/Durham, and possibly Denver as those markets share many characteristics with the ones you have selected
I would start by working with a lender to see the current loans available to you. If they are recourse or non-recourse, how pricing works on these loans, and what is currently available in the market. Like many other things in this economy, the debt markets are changing rapidly with low interest rates but additional reserves required.
I work with brokers on large multifamily transactions (100+ units), so this advice might be skewed compared to 20+ units that you are looking at (est. $10M purchase price in those markets). In the 100+ unit space, buying brokers are rare, there are only selling brokers and then companies that do all of the underwriting themselves (aka me, an acquisition analyst). Working with only one broker would be a mistake, as there are typically 2-3 big brokers in any given city. CBRE, JLL, NGKF, and Colliers in each of these markets would be a great starting point. It is true that commercial brokers are tougher to approach, they will want to see that you are really serious in buying before they spend time on you. It is important that you show them that you are ready and able to close on a transaction when they have one for you.
If you are working with a selling broker, RUN ALL OF YOUR OWN NUMBERS! Learn how to run the numbers yourself, or hire someone to help you that has experience. Selling brokers will inflate the current NOI and will get to a cap rate that is 50-100 bps higher than the real cap rate. Especially getting into new markets, you need to learn how real estate taxes are calculated on the sale, learn what current market rents are in the area as well as current operating expenses for the comps, learn current cap rates.