helaeon

Not really seeing a huge problem with your rocket. I'd just move the SRBs down so all your first stage rocket motors are even. TWR off the pad might be high, so you might want to just keep your throttle down. I usually want to take off at a TWR of about 1.6 and start the gravity turn about 100m/s

With high gimbal engines, sometimes the gimbals point the wrong way when added in symmetry (happens on low gimbal too, but it's not as big of a deal). You might want to add them on one at a time. I'm also seeing too much control authority there. Many of the vibrations are caused by the vectors occillating back and forth setting up a harmonic. Might want to turn down the gimbal range.
Some aero fins are probably a good idea too. Not 100% necessary but I find they can make a big difference, and don't have a huge loss of Delta-V when they don't. I'd put those on the core stage, and two on the drop tanks.
Location of your root part matters too. I've built space stations with their root part far away from the SAS and on the edge of the craft and it vibrates itself apart. If I move the root closer to the center it's better. It's also better if your SAS modules are near the center of the craft as well.
Definitely autostrut across any docking ports or decouplers.

Those LV-Ns should be moved out away from the core of the ship just so those nozzles are clear before decoupling that rhino stage (I'm assuming the reason you did that is because that's not meant to be staged right away, or is meant to be replaced later. If you're not making a replaceable stage use a normal decoupler rather than the seniors)

This thing I'd be using autostruts across any docking ports, and on those boosters and drop tanks. the swaying is from the payload in the fairing, so autostrut that for sure.
For flight I'd use that docking port at the very top as my control point. You just draw the camera inside the fairing and do "control from here" I don't know of a way to change it in the VAB.

Hope that helps Doing the above just real quick I was able to fly it to a 150x150 orbit without too much problem. Little wobbly at the start but then  it's fine. Ended on the Rhino stage with about 1100m/s delta-V