Is there a way to connect the two beams in this angle and still keep the connection strong and keep the rotation the same?
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You probably could use this, which looks exactly like it's the missing part of your puzzle:
You'll be interested to know that this part has intitially been proposed by the first four fans (MUPs, for LEGO Mindstorms Users Panel) working on the very first NXT kit and regularly ran into the same problem as yours. A lot of Technic fans now name the part after Steve Hassenplug, as Brian Davis started called it "hassenpin", eventhough according to him it was more a collaborative effort. It's nice to see that LEGO does listen to the people it invites to workshops (the big wheels in the Emerald Night are another example of this).
If you don't have any of the 90-degree pin connectors, here is another solution using more common parts. It does add to the profile, but is sturdy.