Firstly, 4 months is on the quick side or at least average. It doesn't sound like they've been dragging their heels. Is there a chain? Also, whilst you are chain free, how do you know that these other sellers aren't and you might be waiting many more months?
Secondly, the sellers of the "new" house will have it into their EA agreement that they have to pay the EA commission even if you buy it outside of the EA (which they presumably don't want, hence being with the EA) as they "introduced" you via Right Move. The EA will presumably get a higher commission on the new house if it's more expensive as it tends to be % based.
Thirdly, I wouldn't think much of you if I were the sellers of either house. If I were the sellers of the "new" house I'd be thinking you clearly are one for games (particularly with a low, under the table offer and reluctance to involve the EA) as otherwise you'd be upfront with the EA and your current sellers by pulling out of buying the other property. Similarly, if I were the original seller and you did pull out at the Nth hour then I'd be seeing how much of the money I'd laid out on my new property I could claw back (likely to be more than you will loose unless they're downsizing or moving to a cheaper area) as would the people higher up the hypothetical chain.
I know you've said that you "want to do what's right for your family" but presumably there was something that you liked about the initial house etc to make you put in the offer and get to this stage? If you didn't like it then you haven't really done what's right for your family either (given you cite needing to move schools, away from friends/family for the original house which is why you now want the other house) and you've given more ammunition to the urban myth that all first time buyers are flakey, indecisive and likely to pull out... which is clearly untrue, but you also wouldn't be painting a different picture.
Just be honest with the real life people that you are currently buying from - if you no longer want to buy their house then let them know so they can find someone else who does want it.