how should you resign?


The conventional ways to resign are to stop the clock and/or lay your king flat. Then you note 0-1 or 1-0 on your score sheet and sign it.
The conventional way to offer a draw is to make your move, then gesture with two crossed index fingers towards your opponent, then press your clock, then note your move with the added symbol (=) on your score sheet.

If your position is super lost, then yeah, it's enough to just offer a handshake. The other player will assume you're resigning.
Also because natural body language i.e. you're not going to look very happy when you do it.
I've seen pros stop the clock before offering the handshake... I wonder what the rules are for FIDE, in any case, I don't stop the clock unless I'm claiming a draw (or making some rule violation claim)... which is probably the more important question/answer. For example if the position is repeated 3 times, it's not a draw until you claim it... and you don't claim it by asking your opponent for a draw. You claim it by stopping the clock and saying the game is over as a draw, and at that point your opponent can either agree with you or argue to the director that you're wrong and ask them to make the game continue.

You only offer a draw after you move. Your opponent does not have to say anything to decline the offer, all they have to do is move and it is automatically declined. I wouldn't personally put out my hand if I offered a draw.

I usually tip my king and handshake no matter the position, and that works fine, but the tournaments I usually go to aren't large upscale ones where people come from all across the country just to play so I don't know if thats the proper way to do it. But honestly I think that everything that has been said already is fine, although imo I would tip the king(doesn't matter what order stopping clock or handshake first is fine) just so your opponent fully understands.