Drizzling is pronounced /ˈdrɪz.lɪŋ/, with primary stress on the first syllable. No verified difference between US and UK pronunciations has been found for this word, so the same transcription is provisionally used for both; any variety-specific difference would need to be checked against a major dictionary before being stated. (The example sentence "The rain kept drizzling all afternoon" appears in the following section.)
The stress stays on the first syllable, just as in the base word drizzle.
The rain kept drizzling all afternoon.
Say the sounds
Press your tongue tip behind your top teeth for d, then round your lips slightly as you glide into r.
Say a short, clipped ih vowel, never a long ee, and buzz zz straight after it.
Finish by humming through your nose with your tongue tip down for ng. Let the second syllable stay short and unstressed.
Contrast check: end on ng, not n. If your tongue tip presses up at the end, you have drifted toward drizzlin.
A phrase to say out loud
Carry the phrase into a question form and keep the stress on driz: Is it still drizzling outside?
Try the contrast pair drizzling and dazzling. Both start with d, but drizzling opens with a drih sound while dazzling opens with a flat da. Alternate them slowly to feel the vowel difference.
Test the stress: say DRIZzling, then driz-ZLING, and notice which version sounds like the base word drizzle. The first one is the target.