If you're renting for 2k a month, why buy a house? A $600k mortgage at 3.75% is still $2,778.69 a month payment. It'll probably be over 5% by the time you buy it, which is $3,220.93 a month. That's before taxes, insurance, & upkeep. A conservative number for those would be 1.25% for taxes, 2% for upkeep, and I'll completely guess 200 a month for insurance. That comes out to an additional $2231.25 a month in costs. So your house is going to cost $5452.18 a month. You're currently paying 24k in rent and 20k in savings for a grand total of $44k or $3666 a month. You either need to start saving more or keep on renting. Invest the 20k a year in stocks. If the market is down, don't buy the house.
Really, I would recommend you move or keep renting. If your kids mom is nearby, you could buy yourself and her a house in the suburbs for cheaper than this $750k house you're coming up with.