The answer is... it depends. It depends on how much risk you are willing to take, if you are only comfortable with managing properties that you own free and clear, then that is the right strategy for you. As others have said you will get less return on your money and prolong the process, but your strategy will still work! I am kind of like you I don't like to have myself over-leveraged, I bought a trailer park and duplex in 2017 with 11 doors in total, currently looking at around $460k in mortgages, with about $90k in liquid savings and that is about as stretched as I want to be. I never want to have less than that in savings, because in the event of a huge market turn I want to be able to bail myself out and survive. My goal for 2018 is to pay off both mortgages and search for more properties to buy at the end of the year or early 2019.
If I were you I would find out the 2008 price levels of your properties, where did they bottom out and pay down your mortgages till that amount remains, then buy more. In the event of catastrophic market conditions, you would still be even with your property value/mortgage. That is a very conservative approach in my opinion.
Hope this helps and good luck!