The first thing you have to realize is that the hiring apparatus in the company is bureaucratic. They aren’t scared, mad, judgmental, whatever. They are going to be thinking more about what’s for lunch and whether they can cut out early next Friday than about your candidate profile or negotiation.
As long as you are polite but firm, they aren’t likely to think anything. They’ll just figure they either like your profile and want to work with you, or they’ll figure they know they can’t meet your requests.
Always present flexibility even if you aren’t actually flexible. In theory, if they cannot give a severance package, maybe they can give a much higher sign on bonus, or something else you want. Let them know what’s important to you, but that if there are other ways to address what you’re looking for, you’re open to hear it and think about it.
Once you’re at the stage of making concrete requests, don’t be vague and don’t accept vague alternatives. Always ask for concrete alternatives and once it is stated, always take time to think about it offline, always. That way if you decide you’re not actually flexible, you have breathing room to process your decision and respond.
You should also project confidence about your worth and why you are asking for something.
For example, for negotiating a severance package, you should be very clear about it. In my case, that mattered to me because I was relocating to a new area and at the same time I was switching from individual contributor to manager. I felt the risk of the local job market for an inexperienced manager was high, so if the company I was joining would have restructuring or sudden cuts and I am laid off, the money to float myself in the new region would be high. To feel comfortable about this, I just wanted to know for sure if that kind of change was coming, I would have X months of salary as a cushion.
If a recruiter or hiring manager is too immature to appreciate this as a sincere concern / request of a candidate, and would punish me just for asking either by acting like severance is a taboo way to protect insecurities of being fired or acting like I’m a prima donna, well that makes the decision to walk away pretty obvious for me.
The main thing is just remember they don’t owe you any special features in a job offer and they absolutely won’t offer them unless you ask and make it clear it matters to you.
But you also don’t owe them anything either, certainly not any expectation about being “too fussy” or “scaring” them. Nobody’s going to look out for what you want as a candidate except you.
As long as you’re polite but firm, and you make clear asks and require clear commitments, you should feel completely confident asking for anything you want. Whether you’re willing to compromise or you need to say “no,” you’ll be doing yourself a big favor.