Two months is nothing in academic job terms. If you were offered a job tomorrow, I'd wager that it would take at least 6 weeks before you actually got to start it, paperwork being what it is.
Nobody said you couldn't apply for jobs while still funded, simply that you can't start a job and still be in receipt of your studentship without it causing issues.
Depending on what field you are in, I'd actually recommend starting to apply for jobs anything up to 6 months before submission, given how rare they are in some fields, and the odds of actually getting one.