Every time I try to dial out it gives me a fast busy signal?

Look for settings like local area codes. It could be appending a 1 where it shouldn't. Or missing one. Is that busy a "fast busy" or 120 per minute? that is a mis-dial indication.

View topic - Fast Busy Signal when Calling out - Ooma

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May 15, 2012 - I'm out of town, but yesterday I tried calling a few numbers and alwaysgot the fast busy. Incoming calls were fine. I rebooted the telo and it still ... Can't Dial Out To Same Area Code
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Since you mentioned that you have an outside line, do you have to use a code to dial out normally? Try plugging in a phone to the line for your fax machine, and attempt to dial the number for the other fax machine using the phone. You should reach the fax and hear a high pitched sequence of tones. This is the other fax machine trying to establish a connection. If you still get a busy signal, try calling the owner and letting them know that their fax machine is not accepting incoming calls.

dial those numbers from a regular phone and listen for fax tones. If you hear a busy signal, then they are busy.

This could be related to the fax machine, but more than likely it's a phone line issue. Try sending a fax manually by using the fax machine's speaker or "on-hook", then manually dial the number. Listen carefully as the machine dials and make sure it's even dialing out. Do not hit start until you hear the other fax machine pick up. If the "fast" busy signal occurs before dialing is complete or right after, have your phone line checked.

DO YOU HAVE DIAL TONE ON THE PHONE? IF YES, YOU MAY HAVE A BAD PHONE BOARD. MAY NEED THE HELP OF A SERVICE CENTER. BROTHER TECH

It sounds like the modem is bad a possible electric serge

680 - There is no dial tone.

All versions of Windows are affected by this error.
No dial tone: There are many possible causes of a No Dial Tone Error.
First, try testing the modem's ability to detect dial tone and to dial by using HyperTerminal with the direct to COM port connect method - see Using HyperTerminal. If you give your modem an AT&FX1 command, and then a dial command, dial-tone detection will be disabled; if you still do not get dial tone, the modem may not be installed correctly (especially if it is a PCI or soft-modem); or the cable from the modem to the phone-line jack may be bad or installed in the wrong connector on the modem - try another line cord, and make sure you are plugged in to the LINE - not phone - jack on the modem. Plug a regular phone into the phone jack on the modem and make sure it has dial tone. The modem may also be defective. Many new modems come with an electronic circuit instead of the traditional electro-mechanical relay used to take the modem off-hook. Some of these designs are proving less reliable; often, they will be 'stuck' in the off-hook position, so unplugging, the plugging in the phone cord may give dial-tone. If this is the case, the modem needs repair or replacement.
If the modem will blind dial (x1), but not detect dial-tone otherwise, the problem is likely to be in the proper selection of the Country in Windows->ControlPanel->Telephony or in improper installation of the modem or lack of support for your country in the driver you are using. Check to see if there are updated drivers for your modem.
If the modem dials properly with detection (x3) in HyperTerminal, but does not detect dial tone when using DUNs, the problem may be DUNs (check for upgrade or re-install) or the proper setup and modem selection for the DUNs connectoid.
For Embarq telephone service customers, this error can indicate that you have a message waiting in your voicemail. Embarq changes the dial-tone sound to a "fast" busy signal when a new message is present in your voicemail box. This "fast" busy signal can confuse the modem into thinking there is no-dial tone present on the line.

busy signal can mean another phone on your line is till connected. remove each phone one by one till the line is open, this can be also a redial on a phone activating by itself, or if you have forward messaging activated it can do this as well if it stuck on.

of course..is that phone line, by chance a high speed internet connection?