No need to be embarrassed about college savings. The very vast majority of families have no college savings…or if they have college savings, it’s not very much.
As an example, the Thumper family didn’t have a nickel of college savings. Zero. But both parents were professionals employed full time. One income was used to pay family expenses. The other was dedicated solely to college costs for seven years. For us, that plan worked just fine as the one parent income was sufficient to pay the bills. And the other was sufficient to pay all of the college costs at pricey private colleges for seven years out of current income only.
You are so…so fortunate to have the many SUNY colleges, and CUNY as instate modestly priced options. Once your kiddo has ACT or SAT scores, it will be easier to point you in the direction of finding potential merit aid which could make college affordable.
I would suggest that you formulate a budget for college…in other words, how much can you dedicate to college costs annually for your kid. That number will be important at some point.