You can change both the cassette and crankset without having to change your rear derailleur.
First, not being able to climb well with a 53-39 crankset and a 11-23 cassette is not a problem. That's what the top-level pros race on. You're not that strong - almost no one is. And even some pros don't use gears that big. Alberto Contador used to do hilly segments with a 50-34 crankset and an 11-32 cassette.
And unless you can spin out a 53-11 combination (meaning you have pro-level sprint capabilities...), a 53 is wasted. You don't sprint at 40 mph/60 kph, do you? And for any downhill where your spinning out a 50-11 combination, you'd do better tucking and getting as aero as you can.
Change your crankset to a 50/34 compact and your cassette to an 11-28. Your rear derailleur is rated for 27-tooth max cassette size, but this is Shimano. A 28-tooth cassette will work.
There are a lot of options for a 50-34 crankset. Assuming you have a Hollowtech II bottom bracket, you have a wide selection. And you can almost certainly use a 10-speed crankset such as a 105 FC-5750, or a new Tiagra R460. Or you can get a used 9-speed 50-34 crankset to match what's already on your bike.
A 39/23 combination results in about 44.7 gear-inches, while a 34/28 combination results in 32.0 gear inches.
It won't get you as low a gear as changing your rear derailleur would, but it might be enough. And if it is, you won't wind up with some of the really large gaps in gearing that an 11-32 or 11-36 9-speed cassette has. And if it's not enough, you can change out your RD and get a larger cassette later.