Your conjecture is indeed quite intriguing, however one aspect of it plagues me with doubt as to its usefulness. The need to "play around" with different exponents that are picked at random leaves quite a lot of room for error. I believe there must be some way to relate how far one must take the exponents with the value of the prime that determines the set of primes you use.
What I am trying to say is that if we define the exponent of all primes in the set to be y then there must be some relationship between y and (x-1) primes or simply x.
If such a relationship could be discovered then it would greatly improve upon your current method of arbitrarily picking exponents and would thus go a long way in proving that your theorem can indeed solve all the primes between x and x^2.
x = 13
set of all primes < x^2
abs( 7^y * 11^y +/- 2^y * 3^y * 5^y)
y = some relationship to x
I hope you can see the usefulness in this contribution. If such a relationship between y and x could be discovered, your theorem would be greatly improved.
Hope this helps,