Paul talks about rising with a different body. I think he means that it is our physical body which shall literally be changed. I believe in a physical resurrection from the dead.
Jesus rose physically. It was His same physical body which rose from the dead. And Christ is the firstfruits from the dead. He is our forerunner. Our resurrection shall be after the same pattern as His. A physical resurrection from the dead.
Jesus said all who are in the graves will hear his voice. It's our body which is in the grave - not our spirit. It is our body which shall come forth.
If the spiritual body to which Paul refers is something which the dead in Christ already have in heaven, then why would the dead in Christ rise first, at His coming? They would already be risen, if the spiritual body is something they already have. But no, the dead in Christ shall indeed rise first, because it is their body which shall rise. Then we who are alive and remain shall be caught up together with them in the clouds. We shall all - both the dead and the living - be changed. Yes we shall all get our spiritual body. But it is our physical body which shall be changed. Therefore the dead in Christ need to rise first - so they can recover their physical body - so it can be changed like our body shall be who are still alive and remain.
Even the unrighteous shall experience a bodily resurrection. Only for them it shall be the resurrection unto damnation. But for us - it is a resurrection unto life.