@Megha *******
you are very lucky that the visa doesn't have the name of the employer written on it in thai. They usually do that and it would mean that you can't use it for another employer
What you do now is get the documents from your new employer and go to the immigration office to apply for an extension of stay based on employment. You employer will know the extension documents you need.
you need to meet the requirements to get the extension with documents from the Ministry of Labour (your work permit from the NEW company) and from your new employer.
That's who you need to be talking to not the labour dept, not immigrations, not your embassy, but your new employer