I can understand why it's important to have your welcome email send out to your contacts. When activating a welcome email in your account, that email will only send out to contacts who have signed up using a sign up form created by either Constant Contact or OptinMonster.
Although you can certainly use a sign up form created by other 3rd party developers to bring contacts from your website into your Constant Contact Account, the source of these contacts will display as coming from "Other App" rather than "Sign Up Form". In order for a welcome email to send to your contacts, they will have "Sign Up form" as their source. You can find more about different contact sources and what they mean here.
If you plan on continuing to use a 3rd party sign up form, I'd recommend you look into email automation. This is a great alternative to using a welcome email, you can set up a trigger for contacts who have recently joined. More information on how to set up an automated series can be found here.
I apologize that your welcome email is still not sending. Are you trying to sign up on our forms on your website with a new email address that is not in the account already? If you are not, the welcome email will not trigger to that email address.
Thank you for providing me with that link. I did a test with my email address and I was successfully added to your contact list, but I also did not receive the welcome newsletter. Can you please email social_support(at)constantcontact(dot)com your username, a reference to this post, your best contact information, and a link to your sign up form and we will be able to do some more research on this for you. Thank you!