I thought Aleesha Dixon came from Hertfordshire, like Victoria Beckham. Katie Price speaks in the same way and she originates from Sussex, doesn't she? There are WAGs a-plenty who live in Surrey too, so I don't think it's purely a mode of speaking that's confined to Essex, just that Essex has become more parodied for it!
I agree, it is an exasperating trait. 
Perhaps Alan Sugar and Aleesha would feel as though they were betraying their backgrounds by changing the way they speak - they're financially independent and have been selected to appear on 'telly', so that may be all the justification they need for remaining as they are, if the thought even occurs to them.
Jilly Cooper, Maggie Smith and Helen Mirren all came from Essex and escaped the prevalence of glottal stops and orangeness, however, so perhaps upbringing, decent schooling and motivation, or the lack of it, have a greater influence than the county itself, regardless of the sterotypes. One of my aunts lives on the Essex/Suffolk border and jokes that she has, so far, resisted becoming a 'we wozzer'. 
OP, the way I deal with such irritation, is to switch the whole lot off! Nobody would be influenced to change the way they speak, soley by watching Aleesha Dixon & Alan Sugar, in the same way that those who already speak that way wouldn't be influenced by the RP of newsreaders all that much. Would Alan Sugar and Aleesha even be on our screens if they had had middle class upbringings? A well spoken, Berkshire bred entrepreneur, who attended an independent school just wouldn't evoke the same curiosity as the 'self-made, rags to riches businessman', would it? That's why Peter Jones isn't presenting the show and is only on a panel of Dragon's Den. The pantomime rolls on!