I am having the same problem with my '03 Escape's alternator. I've got a voltmeter plugged into the cigarette lighter socket and can see it's charging OK when the battery check light is off and running on the battery when the light comes on so at least I know the check light is working correctly.
The check light goes off when I accelerate hard so I am thinking that either the power leads are loose or maybe the whole darn alternator is loose. But I'm not used to working on this thing (used to be the wife's car until we bought her an Explorer to use) and I don't know where the heck the alternator is hiding.
I called a mechanic I use and he mentioned that he replaces the alternators frequently on Escapes and the Mazda look-alike because they tend to overheat and fry some components in the unit. Anyone ever heard of that? If true there must be some way of getting cooling air down on it (wherever it is...). He also mentioned that he has to pull the front drive axle to access the alternator and the whole job takes about 3.5 hours to replace it. What? May I please have a word or two with the Ford engineer who came up with that arrangement?

Good luck to all of you with this problem!

Jack
'79 F250 360V8 (alternator easliy accessible)
'90 E150 300 (alternator accessible with some difficutly)
'95 F350 7.3L Powerstroke (alternator right on top, thankyou)
'03 Escape V6 (haven't found darn the alternator yet!)
'08 Explorer (wifey's car - no need to look - that's what AAA is for)