Patient, in numbers: the prescription azelaic acid label for acne cites improvement "within four weeks" (https://dailymed.nlm.nih.gov/dailymed/fda/fdaDrugXsl.cfm?setid=ebdfcaf7-b5d6-47e1-be87-ccaba4b97e75), and the rosacea trials for 15% azelaic gel measured their endpoints at 12 weeks (https://dailymed.nlm.nih.gov/dailymed/fda/fdaDrugXsl.cfm?setid=63773349-0295-4d56-a72e-0d7cade88ddb). So two weeks is genuinely too early to call it — the realistic checkpoints are: possibly something by week four, a fair verdict at three months.
The tingling you're getting is common with azelaic acid, tends to fade within minutes as you've noticed, and usually disappears entirely after the first few weeks as skin acclimates. It's not a sign of damage. If you're on once daily now, building to twice daily as tolerated is the standard dosing arc.
Two things will make the trial fairer to the ingredient. First, judge with photos, not memory — same light, same spot, once a week; redness changes are gradual enough that day-to-day comparison hides them. Second, hold the rest of your routine still and gentle: bland cleanser, bland moisturizer, daily sunscreen, no new actives, since UV and irritation both feed the redness you're measuring.
If the three-month photo shows nothing, that's real information, not failure — take it to a dermatologist. Rosacea has several prescription options beyond azelaic (topical and oral), and a derm can also confirm the "rosacea-ish" part of your description, which determines what's actually worth trying next.