It shouldn’t be necessary to cut the valley rafters into the existing roof as long as the existing roof will support the weight of the new one. Even if it won’t it still makes no difference as you will have to beef it up regardless of which method you use. Most roofs built in the last 60 years or so are plenty strong but if in doubt you can add a knee wall in the attic or add some bracing down to a load bearing partition. This is probably not necessary however. Now to answer your question: First build your roof over the addition but don’t sheet it as yet. Use a string line or a lazer to transfer the top of the ridge and it’s centerline to the existing roof. Mark this point and measure between it and the last set of common rafters/closest truss to the existing roof. Cut a ridge board out of appropriate lumber and place it according to your line/marks. I should mention that you need to snap lines from the mark that you transfered with your line down to the points on either side of the roof on the new addition which will be the start of the valleys. It helps to pull a tape from the same spots on both sides of the addition up to the mark designating the end of your ridge(where it dies into the existing house roof) in order to establish that the ridge is square to the existing eaves and straight with the ridgeline on the new addition. This establishes your valleys. I am assuming in this discussion that you have some idea how to cut rafters and have some basic framing knowledge. Lay out your rafter spacings on the snapped valley lines and then tape them off from the top of the ridge down to your layout marks. In your case you will have a 4/12 plumb cut at the ridge and a 4/12 level cut at the bottom with the bevel set for a 7/12 which if memory serves me correctly is 30 degrees (you can verify that with your speed square). The rafters are then nailed on to the ridge and to the sheeting and the rafters underneath where possible. If you desire to spread the weight over the rafters then you can install a “california valley” which is basically a 2×10 with the ecge beveled off and nailed on the valley line. The rafters are then nailed to that at the valley end. This is definitely the better way to go but is more work and money.