@Makkun70,
Your visa and your right to live in Hong Kong remain until the expiry of your visa. However, you can not legally work for another employer until your apply for a chance of sponsorship through the immigration department.
The question of whether or not to take the payment now or over 3 months is your decision. People may have different reasons, tax reasons or for resume, etc. However, you need to officially separate before you can apply for a visa for another company.
To apply for a change of sponsor, you must do the following:
1) Receive a new signed contract of new position, start date upon approval of visa, and make it a permanent contract, i.e. no dates. If you have a 1 year contract and 15-16 months left on your visa, I think it could cause an issue, but I don't know.
2) Have new employer fill out the form, ID990B, it's on the immigration website. If your new employer has not sponsored someone before, then they need to include a bunch of documents, but if you do it right, it's easy.
3) You fill out the request for extension form, ID91, it's on the immigration website.
4) Get a copy of a separation letter from your previous employer. Even though your employer will fax a form to immigration noting the separation, you must get a copy from your employer, and include it with your application, or it will be delayed.
5) Make a copy of your HK ID card, or pay $2 for copy at immigration office, and delay your queue place.
5) Take all of the forms to immigration in Wan Chai, 5th floor, once you go up the escalator, go to the 3 sets of windows with the queue to get a ticket, then wait 1-2 hours max, and be nice to the immigration guy, he will collect your paperwork and say here is your reference number, we will mail a letter to you. Yet, they never mail a letter, they will fax a letter to your new employer (make sure the number is on the form) and it says that it is approved and gives the expiry date of your visa, which is the same. There is no need to pick up a new visa or make a second trip. When I did mine, it took 2 days. So, yes they say 4-6 weeks, but I think 1-2 weeks is more logical. 1 week if you do it right, 2 weeks if you miss forms.
When you apply for your next extension, you must do it in person and are not eligible for online extension.
Good luck!