You can try the Stabilise filter but I very much doubt you will get a worthy result (it takes a lot of time to analyze), hypersmooth has some major advantages over shotcut’s simple stabilization which only uses video data.
The metadata (gyro angle) which is used by some other apps (like the one made by gopro, reelsteady (paid only)) is there in the file but shotcut doesn’t interpret it for stabilization - and just to mention anyway, any stabilization done after the camera will lower the quality so if you shot in 4k you’ll have to export in a lower resolution.
As for the fisheye, I used to correct it but nowadays I just never bother unless I’m actually cropping some specific area out (and with a smaller area it’s easier to make it look linear).
If you want to experiment you can use this filter from another user which has more detailed options (it exposes defish0r plugin already in the app).
For better results I suggest you film something very straight (maybe you have a wall or a symmetric bookcase, or even a big monitor with a grid on it), enable the shotcut lines overlay and find the best settings to straighten it.