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I think this may be an issue for an immigration consultant, but you would probably have to undergo the entire process again. This is because, under the Kafala system, your visa and status in the country is dependant on your employer.
You would be unable to go over sponsored by one company and work for another, and the only way you would be able to change jobs is to get your employer's consent which, in many, many cases, is very unlikely.
In my opinion, your choices would either be to stick to this company, which guarantees you entry; or cancel your trip and count on this other company to bring you over, but start the process again.
Have a read through our Visas for Saudi Arabia and Working in Saudi Arabia pages for more information and a good overview of the visa and employment systems in the country.
Best of luck,
Xavier
I think the best thing to do would be to speak to the new company and ask them how to proceed, especially because they will be responsible for the visa process to a large degree. Also, contact your closest Saudi embassy to make sure the information you get is correct. Sorry I can't be of more help, but this is a very specific situation.
I really hope this works out for you.
Best of luck,
Xavier