Hey Monica, here a few things to think about.
First, take an evening dinner cruise on Lake Michigan and look at the Chi-town skyline while "listening" to this melody. Listen for the street corner sax player under the EL playing this tune. Imagine him back in the 20s or early thirty's. Listen hard enough and you will hear him or her. I don't care what anyone says, Chicago has the best skyline for that mood (even over NYC).
The tempo is fine, much faster and it is too much. Gershwin was known to play everything fast.
The instruction does say "poco rubato" this is either a mistake in Italian and he meant "un poco rubato" (a little rubato) or it is an advisory against tooo much rubato. Either way, it recognizes the need for rubato. At the risk of starting a new discussion on that topic, if anything calls for the "Chopinesque rubato" this is it.
Think of the eight notes in the melody as "flexible" -- they can range from purely even, straigth eigths toward a more lilting un-equal "blues" eigths (though, to me they shouldn't be as strong as an eight triplet with the first two tied -- that is more for the middle section with the dotted eigth, 16th pattern, which can be a little lazy). But you might try a little bit of the melody with the strong "blues" eigths just to get the feel and then bring them back toward straight eigths.
Notice in measure 7 the tenuto lines over the C#s. That is an invitation to hold them a tad longer and then shorten the following 8ths appropriately.
Be careful on the quarter note triplet in measure 8. There is again a tenuto line over the last quarter -- don't rush it.
One other thing about the melody. The grace notes that occur from time to time to me represent that saxophone's slide into pitch. They can sometimes be a little longer.
I see the left hand part as containing 2 parts, the bass player and a middle counter melody. There are 2 ways to look at this. First, the bass line is C#G# F# G#... and the counter-melody is E E# C# E#. The second way is that the bass player is playing C# G# C# G# (crossing over the counter-melody), and the counter-melody is E E# F# E#... Either way, the hi C# needs to connect with which ever line you see it as a part of. It sort of sticks out from time to time.
The other issue is, of course, the 10ths. There are 3 physical sizes of 10ths on the piano (as opposed to musical sizes) the small 10th is white to black or black to white m10 (the first 10th in the piece). The medium one is white to white or black to black major or minor 10th (C - E, D - F are physically the same size). The large one is Black to white or white to black Major 10th (D - F# or Bb - D). Personally, I do ok with the first 2 but I can't do the large one. In your case, your issue begins with the small one so there needs to be a work around.
When you have to break them, I would suggest being a little more rhythmically decisive than just trying to get the 2 notes in as fast as possible. What I mean is to make one of the 2 notes a definite anticipation of the beat and the other directly on the beat (the first might be about the last 16th of the previous measure). Also, when breaking things like this, there is no rule that says the bottom note must be first. Experiment with a downward break (E - C#). Try mixing it up -- sometimes upward, sometimes downward. Another possibility is that after the pattern is established, leaving one or the other notes out, just to get away from the constant "ba-dum" might be an option. The idea being that once the listener has heard the pattern enough, they will sub-consciously fill in the blank.
Because of the need to break the initial m10 of the pattern, I, as other's have suggested would use both hands to play the walking 10ths in measure 10 together (even though that has an indication allowing a break).
I hope this gives you some ideas. Since this is Gershwin and based on jazz and blues, IMHO it gives you a little more flexibility to alter things from the written page.