The issue is, as you may know, that there is not one 'standard' Sephardi pronunciation. There were a variety of traditional dialects for different 'edot — Moroccan, Baghdadi, Yemenite, Persian, etc... Then there's the generalized 'Sephardi' accent of modern Israeli Hebrew (à la Ovadia Yosef) and even general Israeli Hebrew which retains some 'Sephardi' features.
So first choose what kind of Hebrew you would like to adopt, then find something to practice! I find what works best for me is listening to recordings of people speaking or singing that I first follow along, then practice speaking with them, then I practice it on my own... Eventually I'm able to internalize the pronunciation and replicate it in other contexts.
For example, if you wanted Moroccan-style pronunciation, Darké Abotenou has a lot of videos of Moroccan prayers and piyyutim (I learnt a beautiful Moroccan sheva' berakhot from one of their videos).
Hazmana laPiyut, though, is by far the most tremendous resource. You can search for a particular piyut (Deror Yiqra, for example, or Yedid Nefesh), and it will pull up a list of recordings organized by 'edah: Morocco, Tunisia, Libya, Turkey, Iraq, Yemen, etc... They are either vintage recordings of hazzanim from those countries, modern recordings of elderly immigrants from those countries, or contemporary arrangements by Israeli piyyut groups, but they are usually very faithful to each particular tradition. So listening to the Yemenite piyyut recordings will definitely be the best place to hear authentic Yemenite pronunciation.
Hope that helps! Good luck!