Speaker Damage/Amp Power/Volume

This is pretty simple, you are playing too loud. 250 watts sounds like a lot of power but it's not, it's only 3db more than 125 watts. An in-wall speaker is limited by it's design, it's not meant to rock the house down. There is a reason that there are really big floor standing loudspeakers. If you want big sound you'll need big speakers.
The only guide for feeding power is, that if it sounds loud, it's probably distorting and you need to turn it down. A good speaker will play really loud but will not distort, you'll just notice you are shouting to talk over the music.
It sounds like your audio designer didn't quite meet your needs. At 91db efficient (in room/per Energy) you are not going to get where you want with this speaker and your amp.
Most in-wall systems are designed specifically to trade sound quality/volume for aesthetics. You might want to re-think your requirements.
Thanx, Russ If I have it wrong, I apologize.

Russ69: The 2.3i's are 40 inch floorstanders with dual 6" woofers, a dome mid, and a dome tweeter. Sensitivity per manual is 90 dB.

Goblue: It is my understanding that the typical "tower" speaker will not be capable of delivering the type of volume that you seem to be interested in. You could consider purchasing a professional grade speaker such as those commonly used to DJ. However, I think most would agree that the SPL you would acheive would come at the cost of sound quality, to put it simply. Otherwise, I think that you will need to go to a much higher price-point to get quality speakers that will gracefully play full-range at extremely loud volumes. Of course, it would follow that you will also need significantly more powerful and expensive amplification - money, money, money.

If you are running your 2.3s full-range, consider integrating a subwoofer or two and easing the burden on your speakers and your amplifier. You may be able to achieve the volume you want and keep your equipment safe.

To answer your second question, I think most would also agree that you can never have too much power. In fact, it is usually better to have an amplifier that can run your speakers with power in "reserve". The most likely culprit of driver damage is insufficient power. In either case, you should listen carefully for signs of distress such as "thinning" of the sound, particualarly on the low end, or distortion. Incidentally, I would think that your amplifier should be more than sufficient to power your 2.3s.

The "Thermal Protection" LED on your Adcom will light when your amplifier has shut down due to overheating. It simply provides explanation as to why the unit has stopped working and lets you know that it needs time to cool down. Hopefully, you have never driven it to that point.

Energy will probably fix it for free.
Also, add a powered sub. Not only will this alleviate a lot of the work for your speakers, it will let you get by with much less power from the amp since bass is where most power is wasted.

Unfortunately, there are no external screws or bolts to tighten--the woofer and surrounding rubber ring are flush to front of cabinet. When I described the situation to Klipsch (now owns Energy), I was told that new bass drivers are available for $280 and that they recommend a shop in western Michigan for installation. Before spending that money and driving 200 miles, I will find a place in suburban Detroit to check the driver. I also will begin to research subs and how to integrate. Regarding sound levels, 95 percent of my listening is at lower to moderate volumes with sound quality being the clear priority. The onset of the buzz, however, has me "gun shy" and resisting the occasional urge to turn it up. Your suggestions are appreciated!

Goblue:

You will find the bolts for the woofers located on the rear of the speaker. You will have two of them, one for each bass driver. They are center mounted. They are easily removed, just be careful not to let them tumble out as you unscrew them - or just try to tighten them and see if that cures your problem, as suggested.