Newsletter and email limit of provider
My advice is to use "MailQ" (For Drupal 7 you can use "Queue Mail" instead) and "Elysia Cron" modules together.

Configure MailQ module to send 50 mails per one batch.

Configure Elysia Cron module to run MailQ cron hook every 15 minutes.
(note you need configure you cron job to run every 5 minutes or even often, more details in Elysia Cron readme.txt)

In this case MailQ will send 50 emails each 15 minutes - and you will got no more than 200 emails per hour.