Help Requested: Video recording with a DSLR or Mirrorless camera.

My Problem: I have an ancient Sony a58 DSLT APS-C camera which shoots brilliant stills but which has serious problems with video.

If I shoot automatically, focus and exposure are adjusted for me but subject or camera motion can result in unpleasant jitter caused by using faster shutter speeds than the traditional 1/50 second (for 25fps footage).

In bright sunlight the automatically selected speed can be faster than 1,1000 second which is disastrous. Shooting indoors where the lighting is lower is OK because only slow shutter speeds can be used.

I have 2 "solutions". External neutral density filters or manual shooting. With my camera shutter priority can only be selected when manual focusing is enabled.

These are OK for planned shots on a tripod but for rapid handheld shooting in variable lighting they are unsuitable for obvious reasons.

The answers are to have a camera with some sort of (automatic?) internal ND filtering system (to permit the use of large apertures) or one which allows fully automatic shooting with shutter priority enabled.

Do any modern DSLR/Mirrorless cameras have these features and if so are there any at a reasonably low price point (sub £1,000).