...sliding the saddle forward will put you more over the pedals, and it might be a good thing on a larger frame. As you have surmised, most people adjust reach by saddle adjustment and stem length. I am an inch or two taller than you, but with a long torso relative to my legs, and long arms. I ride anywhere from a 58 to a 62 here successfully, and manage to do it comfortably by both saddle adjustment and choice of stem.
Flipping the stem might help some, but given your description, I'm guessing you'll need to get a shorter one soon. If you have a bike where the reach is already comfy, measure the distance between bar and saddle post, and remember it for future reference because it helps a lot whenever you start fitting a bike that is new to you. Remember to allow for your riding style and varying your hand position from hoods to bar top to drops, if you're riding a drop bar.
Of the used bikes I've bought, I probably had to change the stem on 70% of them to adjust the fit. If there's a bike co-op in your town, they will have a large box of stems in various sizes and lengths sitting over on a shelf somewhere for cheap.