I suspect you have put the "resolution" field on a screen it should not be on - when the users edit the issue or go through a transition that uses a screen with resolution on it, they are (inadvertently) setting the resolution.
Go over all your screens and remove resolution from all of them, except screens that are used on transitions where you want to set it (i.e open > resolved or in-progress -> closed)
Once you've fixed that problem and stopped damaging your issues, we can move on to fixing the broken ones.
Issues are striked through as soon as they have a resolution. It seems that while transitioning to In Progress a resolution is set via post function. So take a look at the post functions in that transition.Remove that postfunction and possibly add it to some other transition( e.g. to resolved/closed status) in order to set a resolution there.
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