Is "forsought" right in this case?

Provided foreseek is a valid verb (which it appears to be), it will then be conjugated just like any other verb. The correct point of comparison would be seek. You are speaking in the past tense, therefore sought. Foreseek becomes foresought.

You needn't worry about not being able to find dictionary entries for conjugated versions of a verb. If you can find the infinitive you are good to go.

As a comment points out, check your number agreement. Something is singular. Therefore:

They sought something that was their own.

You could also avoid the number question altogether with

They sought something of their own.