There's no real need to move the hands, since it's pretty clear the "OMEGA" logo is wonky, unless there is something really funny about the photos. However, I don't think you'd need to wind the watch in order to move the hands. You would pull the crown out to the "set" position, and move the hands that way. It's helpful to see the inside of the watch to identify the reference, but there are some expert members who might recognize it by eye. I used the word "value", but I could just as easily said "originality", "desirability", or "collectibility".
If it were mine, I would do is have the movement fully serviced by a reputable independent watchmaker with access to Omega parts. And if possible, I'd like to get the applied OMEGA letters straightened out, but obviously that is not absolutely necessary. If the crown is determined to be a replacement (other members will know), it would be nice to have it replaced by a correct crown during service. Definitely don't let anyone polish the case (you should strongly emphasize this to anyone who works on the watch) or refinish the dial.
Other members may have different opinions or advice, so stand by.
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