Instead of your fingering, I would use:
24 - 35 - 24 - 13 - 24 - 3
As for getting the thirds to sound "together", what I would do, and have done when I've encountered third trills, is to practice very slowly, playing slowly enough that each third gets perfect. Something that has helped me is to let gravity do the work - A bit hard to explain, but try placing your fingers on the keys of the third you're going to play, and just relax into the keys, gravity should then do the job. Important not to keep your fingers too relaxed, so that they don't get "uneven" by the resistance in the keys. It's a bit hard to explain, but do some experiments at a very slow pace, see what works best for you.
And play the passage very slowly and perfectly (play as slow as it takes to play it perfect) a couple of times before going to bed.