Im leaving my current job and want to use my sick time but also keep the reference.

I've been at this job for 2 years now. Never been late rarely called out and therfore have a lot of sick hours but I never get sick. I want to ask to use them and if im told no I might extend my last day and call out for a full 2 weeks. How would yall go about this? I want this reference because it'd be a good one for me and my dog as I've been bringing him in since day one.