Your reach is too far. I won't bother with telling you that your top tube is likely too long (the typical small cx bike issue for small women), but, yes, a shorter stem will help. A lot of women are tempted to do the stem-with-a big-rise/angle approach to bring the bars back, and while it technically reduces reach, it isn't the proper fix. It brings you back, slightly, but really brings you up. Unless you have profound neck/back issues, you really shouldn't shoot for a position in which your handlebar is higher than your saddle, imo, unless your bike is designed for it (and yours isn't).

While you can tilt the bars back, this is also not a great solution, as it creates a lousy platform for your hands when you are in the drops (e.g., kinked wrists). The idea mentioned of finding a bar with less drop/curve from the tops to the hoods is a good one.

When fitting yourself to your existing bike, your starting point is your saddle position. Having it in the right fore/aft position and at the proper height is important. It should be considered the baseline. From there, you can start trouble-shooting reach. (You should never move the saddle to fix reach problems, especially if you have any knee problems, although folks do it anyway.)

You mentioned the hoods feeling large. Why Shimano makes HUMONGOUS hoods these days is beyond me. They didn't used to be this way. Shims will bring in your brake levers closer to the bar, but they won't affect how the hood feels. Because you have a Shimano equipped bike, as long as you have a double crank, you can consider SRAM brifters. They are a really nice shape for small hands, without the bulbous look and feel of the Shimano version, and are compatible with Shimano 10-speed drivetrains (with double cranks,, that is).

But, like everyone else said, a professional fit is always a great idea!