I would wait to paint the beams until you have painted the paneling and trim. I have a feeling that they may become really interesting focal points once they don't have to compete with the other wood. I would wait until you paint the rest and then decide.

I agree to wait, however, what look are you going for? What style? What is your furniture, lighting, functionality of the space? All these things play an important part of the design.

Craftsman design. Some antique wooden furniture (buffet, chairs, etc) Chocolate brown leather couch and chair. Blue leather craftsman style chair. Many windows so lots of light now but also installing lots of can lights. We like a comfortable decor, not really fancy.

I like the beams dark because they match the wood accent wall, but I feel like they are weighing the room down. I don't really want to paint the beams, but maybe white wash or gray wash them to lighten them up.

Craftsman style = natural stained wood. If you are replacing carpet with wood floors, painting walls gray and putting in more light, I think it will be nice to keep them stained wood. If you want to lighten room still, do it with your accents - rugs and sofa in a light tweed or linen. A "wash" is nice for some designs but it is definitely not craftsman style. Do a search for "modern craftsman" or just "craftsman" style and pull together blues, darker greens, ivory, taupes and a nice camel tone color story. Get your entire plan for the house together first - color story, wood tones, etc. Then build from that. Take a macro-view first then focus on the micro (little details), otherwise you can end up with a mismatched style that just doesn't feel peaceful.