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In understanding your initial concern, you do not indicate your aperture setting........?

My suspicion is your camera was at or near wide open and you have a high iso range set within your camera.

Under aperture priority fundamentally your camera will in the first instance vary Iso, at your aperture setting,

to achieve suitable exposure/image sharpness within the set iso limits in your camera and min or above SS

speed as necessary.

If your internal iso range settings are  set wide/high then your camera will adjust accordingly above or at your min

shutter speed,  and explains in limiting the SS to 1/1000 and/or in darker conditions(ie reducing ambient

light input) there is still enough latitude in your in camera ISO settings to achieve exposure/sharpness

at your  minimum shutter speed.

Looks like you need to lower your in camera acceptable iso range settings  to address this problem.......also suggest

setting your min speed to 1/1500 as a more typical general SS setting.