I have already put in my 2 week notice for my current employer, as I will be starting another job in about 2-weeks time. However, I failed to remember that I have 4 unused floating holidays/personal days before I already put in my notice. These are 4 days that are granted by my current company at the beginning of each year (no accrual period). These are not paid out upon voluntary termination (unlike regular accrued vacation days that are paid out), as explicitly specified in the company policy. I feel like I'd be losing a huge chunk of time and/or money if I forfeit these 4 days, but at the same time, it's only a week into the new year, and perhaps I'd be cheating my employer if I took them all now? Also, I don't know that my manager would approve this time off in any case, since I'm on my 2-week notice period. Should I try to get these days off, or should I just let it go? Should I talk with HR about my options, or will it come off as scummy and money-grubbing? Any advice/suggestions for course of action on this?
There are occasionally times when it is simpler to let something go, rather than try to grab every last shilling on the table.
This is one of those times.
You can certainly ask your HR department to pay those four days, and it wouldn't hurt to ask them about it, but I wouldn't plan on holding my breath waiting for them to say "yes".
Floating holidays are a pretty new concept that are intended to give employees the opportunity to celebrate the holidays that are most meaningful to them, regardless of culture, religion or nationality. So instead of having 11 fixed holidays you have 7 fixed holidays and 4 floaters (for example).
For standard holidays it is simple: If you quit before the 4th of July, you don't get to take the 4th of July off. It's probably best to view the floaters the same way: as something that's spread out more or less evenly through the year. I'd say, it would be fair to take of one floater for each quarter you have worked through the year. So in your case, that would be nothing (sorry).
You could certainly try to force it, using the letter of the law. But it's clearly not the spirit of the law: If it's getting abused, the company will just tighten up the rules or quit floating holidays all together. It's a well intended and generous policy (by enabling all floaters from day 1). It would be a unfortunate if it would get killed by abuse so the mature & ethical thing to do would be to let it go.
You earned your holidays. Asking for those holidays, or the money equivalent, is absolutely normal. In European countries it's usually not even necessary to ask, because the employer knows they have to pay.
In the USA, rules are different and nastier. It's best to make sure you've taken all your holidays before you give notice. Still, asking HR for payment will not hurt. A decent employer would pay.