Having done plenty of these fixes, I can tell you, each situation is different. Why? Snow Loads.
Your fix is only going to be fixed by an overworried engineer who has to cover their ASSets. Usually an overbuild, but worth every penny for when you want to sleep at night...or sell your house before an Inspector comes a callin.
My opinion is that this house was built back in the 30s or 40s and this was how they did it with no bearing wall ever put in to carry only the joist weight. The whole reason that crack is happening is because of the bad overlap joint at the joist. The walls only have to spread an 1/8th of an inch due to any weight at the roof/ or from wind movement to crack the sheetrock.
Installing a beam in the middle by cutting the joists does not help this issue because of the walls still wanting to spread, so you would add strapping at the bottoms to help connect the joists to each other to prevent this. This is why the beams are usually underneath the joists with a larger overlap at the joists connection.
This happens with roof pitches less than a 6 pitch most often because the roof it to shallow to distribute the weight downward instead of outward.
Installing a beam at the ridge under the rafters is another option to make this a cathedral ceiling. And will need to be carried for the point loads at each end as Neil said. However, so does a flush beam installed at the joists.
Overall, I would suggest calling in the engineer for your solution to YOUR home.Cheapest fix will be a truss type system with lots of metal hardware installed and no guarantee of fixing the issue/ but a pretty good chance of working. My opinion on installing a rafter beam and carrying the weight to the foundation would also install some roof and soffit ventilation. Be careful of this vaulted or cathedral solution because when you sister under the rafters for the insulation, you may not have enough room to the tops of the windows depending on their height. But if you can, you may get the most bang for your buck with the openness and bigger feel to the room. This however also has a downside at first. Any new wood lumber you use will shrink and cause a crack in the new vaulted ceiling.
My advice , call the engineer for 400 bucks... it will be worth it.. may even be less expensive than that.