I would suggest installing a copy of
DadaMail (free) on your shared hosting account, then
invite your forum users to subscribe to your newsletter.
Do not just dump all your forum members' emails in, that would be spam. But if you
invite them, and they double-opt in, then your list is 100% A-OK. (And you know the people on that list are
really interested in your site!)
With Dada Mail you can set any size batching. Set it to
6 emails per minute. Yes, six

that will send out 360 emails per hour ... and leave you 140 emails/hour left over for ongoing forum notifications, etc.

That way you are well under all the limits, but your message makes it out.
Generally when I send out my newsletters, I batch them at 15/minute, and that puts no stress on the server whatsoever ... I like 15/minute because multiple accounts can be sending newsletters simultaneously and the server doesn't struggle with it at all. Both
DadaMail and
1-2-All work great.
I know 15/minute is higher than HostGator will allow; I'm just explaining that that is a reasonable rate for a server that is not overloaded.

Bailey