Dual Citizenship - Passport questions

Hello everybody,

I had little issues while travelling around S-E Asia last year and had two red stamps on my former French passport because I stupidly overstayed twice in Thailand for a couple of days. I then  went to immigration office in Bangkok, paid the fine with a wai and a smile; long story short : no problem at all, they even seemed happy to get these extra bahts.

Now -after more than a year out of the country, I'm back here looking for a job and things are going pretty well, I will definitely get a work permit + visa with the help of a language school - poor salary but hey, I'm not here to make loads of money, just to live another experience in this amazing country.

As I didn't secure a job yet and can't be 100% sure until it's approved...etc. My questions are :

- I've entered Thailand this year with a British Passport (I have dual citizenship FR/UK) : will Immigration be able to find out that I was previously here with a French Passport ? Or am I a totally "new" person in their system ? I had never travelled here with the UK one before and need to extend on 10/11/17 for one more month (that would give me another 30 days to find a nice job...)

- When you get a contract with a school that provides Visa and WP, does it take long to get the proper legal 1 year visa ? Or is it a short process as soon as they sign you in ?

- Would you recommend  to get out of the country on 09/11/17 and do a visa run or to make the extension at the Bangkok Immgration Office ?

THANKS in advance for your advices :) Krop khun krap.
Chok dee krap