How can I tell if I am overdriving the amp or the speakers?

It can also be the room, or speaker/room interactions by the way.  You can experiment with blankets and pillows.  Put some around the room and see if they change that aspect of your listening.  If they do, in a positive way, consider room treatment.

The walls are an obvious choice, but I've found that some of this hash lives on the floor between and behind the speakers.

Your first clue is the sound taking a nose dive.   Becoming hard, brittle, annoying, listener fatigue, etc.   The second clue is any warning LED on your amp.   The third is blown fuses.  Your amp may not be clipping but may be current starving at some frequencies during slewing, which would cause the "sharp" sounds you are hearing.  It is hard to say but you clearly have something going on that is not right.  Note that high capacitance speaker cables can also cause this effect.   Do you have a long run of speaker cable?

start sounding sharp and annoying. 

That is your first warning. But you really should not be playing that loud. I always reduce the volume until I am at a still satisfying loudness and not overdriving the room, the equipment, or my ears. If you are running out of steam...you have to pay to play but 250 watts should really be able to do the job...however 500 wpc will do more. 

Please tell us your front end components and cabling on the front end. Also, give us an idea of speaker and listening positioning. Toe in etc… Thank you.

"The amp is 250 x 2 into 8 ohms," and the speakers are rated at 90?

I wouldn't think it would be the amp running out of power.  I would think the speakers are just trying to work to hard.  But I also think that the room acoustics suggestion has merit.  Back in the late '90s & early '00s, I was in the mode to be pretty much be playing my stuff to loud all the time (& the stuff I was using wasn't cheap stuff, either). I would be jamming out to ear bleed levels in the living room & thinking that I was at the cusp of clipping, but when I'd go out on the back porch to barbecue or whatever, everything sounded sweet & clear with no distortion.