Trouble sending mail after hacking episode

Here's a unique problem: I have two email accounts. Acct 1 is with my little mom-and-pop ISP. Acct 2 is with Gmail. I use Apple Mail for both of them. Last weekend, my Facebook and Gmail accts got hacked. The hackers sent out the typical "I'm stranded in London, please send money" sort of scam. I followed the steps to regain control of my FB and Gmail accts. But ever since then, I have had a problem sending email with Acct 1. I can send and receive just fine with Gmail, and I can receive with Acct 1, but I can't send with Acct 1. I've double-checked all settings, and I've contacted my mom-and-pop ISP several times, but we can't figure it out. At one point, the problem went away for one day. I don't know if that was because of anything I did or not. Then the problem came back. If I log in to Acct 1 via the ISP website, there is no problem sending. It seems odd that this problem could be related to the hacking, but I can't imagine that it isn't, because of the timing. (Not to complicate matters, but a day or so before I was hacked, I began to hear a very faint beep on occasion from my computer. I've heard it maybe six or seven times in the past six days, and I just heard it awhile ago. Just one quick, faint beep. The paranoid part of my brain is wondering if that's a sign of some sort of malware, even though I've read that malware generally isn't a threat to Macs.) Any ideas on how to troubleshoot this? Advice would be much appreciated. Thanks.

Within Apple Mail, for Acct 1, are you using that ISP's incoming and outgoing mail servers? And for Acct 2, are you using the Gmail incoming/outgoing mail servers?

Have you checked that outgoing mail server for Acct 1 is properly confirmed per the ISP? The port, need for SSL, need for authentication?

Regards

Yes, for Acct 1, I am using my ISP's incoming/outgoing servers, and have been on the phone with them to ensure that all settings are correct. The tech guy at my ISP is stumped, and so am I.

For Acct 2 (Gmail), I am using the Gmail incoming/outgoing servers.

No settings have been changed since the hacking episode.

Thanks.

For Acct 1, are you using the ISP provided email address? I probe because I use Comcast as my ISP and used a third-party email address but send emails through Comcast's outgoing server..this used to work for a long time with no authentication needed..

However, about a month ago something changed and now I have to use my Comcast username/password to authenticate to the outgoing server to continue sending the email from my third-party address..

Do you get any specific error when you try to send the emails, that might help narrow down to what might be going wrong...

Oh and the hacking episode, though it might've happened at the same time, wouldn't cause the outgoing mail problem..

Regards

Yes, I'm using the ISP-provided email address, which I have been using for about five years.

The error message simply says it can't send the message using that server (my ISP's server).

I appreciate your attempts to help me figure this out.