if they prescribed you meclizine, i don't think it will help with any anxiety-but i'm not a dr
i personally don't like it, it doesn't help me at all with my dizziness, and it actually has an excitable effect on me-doesn't make me calm, but i'm kinda weird with medication. i was prescribed xanax for my anxiety-which i've never had until i got this vertigo. i've been taking it for about a week now, and it has helped my heart racing, nervousness tremendously! i take half of one when i get up, and the other before bed. before it, i wouldn't be able to go to sleep until about 4 or 5 in the morning, and wake up constantly with my heart racing, etc. i can now sleep through the night peacefully. only thing, it kinda makes me a little "slow" during the day, which i don't really care as long as i'm not all freakin' out. maybe check with your doctor and let him/her know you are having this anxiety.
but the antivert is worth a shot for your dizziness, so try it out and see if it does anything for that. but i highly doubt it will help at all with anxiety
hope you get better fast, along with the rest of us! good luck