Nervous about giving in my resignation at work

So then tomorrow you don't resign, but explain to your boss that you've applied for another job and they will be receiving a reference request from the company you've applied to. You do not need to tell your boss at this stage that you've been offered the job or that you've accepted it, as the offer is contingent on your references being acceptable. Once your prospective new employer has confirmed your references are acceptable and forwarded your contract for signing, then you can resign. (NB: It is commonplace these days for employers to provide the most basic of references which merely confirm your job title and your dates of service. This is because employers can be sued if they provide an inaccurate or discriminatory reference so many no longer give detailed references.)
Changing jobs throughout your working life is not unusual and your boss should respond professionally even if they're not thrilled about the additional manpower shortage (that's their problem, not yours). Furthermore, if they want you to do a conscientious, diligent handover to your colleague, it would not be in their interest to make your final weeks of work there uncomfortable/unpleasant.
If your boss asks why you've looked elsewhere simply tell them the truth (or something like "Well, this new job offers greater development and advancement opportunities and a higher salary.")
Re. your annual leave, you can certainly ask if you can leave earlier than your notice period by taking any unused A/L, but your employer is within their rights to refuse this request due to business need (which seems quite likely given that you say they're already short-staffed).
Finally, you say you still have 21 days holiday left, but unless you've carried over a lot of unused leave from last year, I think what you mean is that this is what you would be entitled to if you worked for your current employer for the rest of the current leave year. Obviously, you're not planning to do this, so your annual leave entitlement will be pro-rated to the length of time you do work there in the current leave year. A simple example: if your A/L entitlement is 28 days (inc. Bank Hol's) this would be 2.33 days per month. If you work there for 5 months of the current leave year, your entitlement for that period would be 11.66 days. If, for example, you've already taken 5 days A/L, you would be paid out for the remaining untaken 6.66 days (or you could leave 6.66 days earlier if your employer agrees to your request to do so).
Best of luck for the future OP. 😊
PS: I've mostly only read your posts, so sorry if I've repeated what PPs have advised.

You’ve had some good advice on this thread but I don’t think you appreciate that if you resign before a formal job offer, you’re at risk. A subject to references offer is not a formal offer.
I would tell your boss tomorrow you have been offered another role, subject to references. Discuss with him/her your likely end date, given you will likely resign at some point this week, when your new employer receives your references.
Formally resign once receive contract.
this is pretty standard across all industries.
you get annual leave pro rata to the amount of the year you’ve worked. If your annual leave entitlement is 24 days, you get 2 days per month you’ve worked. That means if you’re annual leave leave after say 2 months of the leave year, you’ll get 4 days. If you’ve taken more than that then they’ll reduce your final pay to reflect you owe them leave.

hope that helps make it clear. Good luck!

I put you’re not being unreasonable because LOTS of people dread handing in their notice for the exact same reasons! I’ve been there myself and was SO anxious about it and the “repercussions” (there were none!) because it feels like you’ve betrayed your current company (you haven’t!)… however my boss at the time was nothing but supportive, but sad to see me go.
Professionally they can’t be funny with you as it’s your life and your choice. The lack is staffing isn’t your responsibility.

If your boss is a professional, they’ll accept your resignation with respect and well wishes.

Can’t you give references of your previous employers? That way you can fully complete the reference process + contract signing before you formally resign.

Echoing the advice not to resign without signing the contract first. Any offer is contingent on references anyway, so it makes no difference to them whether you have signed or not.

When you've signed, tell your boss and then follow up in writing.

You won't have 21 days of leave left to take as that's your remaining entitlement based on you working there for the rest of the holiday year. As you are leaving they will calculate your pro-rata holiday entitlement. You can ask if you can take this as part of your notice period, but they are allowed to refuse this - particularly if it will leave them short staffed.

So wait until they're sorted. Presuming you've never been so infuriatingly obtuse and slow to follow simple information in front of your boss, and they have something positive to say.

Please resign in person. It’s so very rude to just drop an email.

then follow up with an email.

its fine to ask about annual leave. It’s also fine for them to decline. If you can use it they will pay you for it.

Both, arranged a quick meeting with your manager, resign verbally. Follow up to confirm conversation and leaving date in writing.

Do not resign until you have a signed contract.

Tomorrow ask your boss to use them as a reference. Wait for confirmation of contract, get it signed and then resign.

And no doubt the offer letter contains wording along the lines of "This offer is subject to acceptable references"?

It's pretty standard for companies to wait until they have references before issuing a contract. As long as you have an offer in writing, it's fine. And yes, your leave will be pro rated, but you may not be able to take it as part of your notice if you have a lot to hand over. It's discretionary, as with any leave. You'll be paid for it if you don't take it.

You can put in references on Monday and not formally hand in your notice until you’ve signed a contract, your start date at the new job will just be later

Then I refer you to my earlier post - the only conversation to have with your boss for now is please will you be my referee?

Oh my god, that's all a chain of events that haven't happened. I'm sure they will, but you resign after they receive references and everything is signed.

But do what you like. You seem to think you're providing your own references...

so you’ve not given them a list of references yet?

So don't resign yet.

I'd wait until contract signed to put notice in.

But you won't have that much left if you're leaving next month.