First, to clarify, a Dyson Swarm is a cloud of orbitals around a star to use most or all of its energy via solar collectors. This was originally called a Dyson Sphere but was renamed because of confusion: people thought a Dyson Swarm was a rigid shell. It never was.
Anyway, the classic orbital is an O'Neil Cylinder, which would be 3-4 miles in diameter and 10-20 miles long. You spin it to get earthlike gravity. With this diameter the rotation isn't likely disconcerting and the centrifugal forces aren't so large as to tear it apart.
This kind of structure could be built with a material no stronger than stainless steel and using solar panels for power. It's relatively low tech. There aren't any exotic physics or exotic static states of matter required. It's basically an engineering problem and can be built incrementally.
Why do I mention this? Because fo rthe distances involved, an interstellar starshhip is basically just an O'Neil Cylinder. You need to support people for centuries. Such a cylinder could get 10s of thousands of people, possibly 100k+ to another system in relative comfort.
So how do you get to another star? The tyranny of the rocket equation means any form of propulsion where you need to carry to propellant just won't work with the possible exception of nuclear fusion.
But what if you didn't have to carry propellant at all? To accelerate or decelerate. That makes it way easier. But how would yo udo that? Easy, at least in theory: solar sails. The solar wind carries pretty significant momentum. A sufficiently large solar sail (and it would have to be large for such a ship) would absolutely be capable of accelerating a ship to at least 0.01c. And you can decelerate with the same solar sail.
You can do even better by collecting energy from the Sun and concentrating it on the sails, which is yet another reason to build a Dyson Swarm.
The energy budget to travel to even our closest star is so vast that we would need to do things like collect most of the Sun's output energy. I personally don't believe FTL is possible. Time dilation only really kicks in meaningfully at >0.99c and I just don't think that's parctical and, if it were, the energy required is even more vast.
In fact, at 0.99c you would suffer drag from the interstellar medium (gas and dust)..
So any intersteallar ship is going to be a generation ship, a habitat.