Landline beeping issue

If you are getting stutter dial tone when picking up the phone that means you have a voicemail message from the phone companies voicemail service. Are you saying you're checking the phone companies voicemail or your phones systems answering machine?

I was answering my phone systems answering machine

If it's coming from your answering machine what's the make and model? You said when you picked up your handset you heard it. You can see what stutter dial tone sounds like on this video if the mods will keep access to it. 

It is coming from the phone when you try and dial out

I followed the instructions given and it does not stop

You followed what instructions? What's the make and model of phone? Do you get stutter dial tone as shown in the link? Does the stutter dial tone stop and go to regular solid dial tone after a few seconds? Who is your carrier?

The make is AT & T not sure of the model.  Yes it stops and goes to a solid dial tone.  Carrier is Xfinity

Yes it stops and goes to a solid dial tone

Then it's not coming from the phone it's coming from your phone carrier. Have you checked their voicemail to see if you have messages. Just AT&T without model is no help, there's hundreds of models, which is immaterial since you now identified the "beeping" as coming from your carrier.