I have a 2-line phone AT&T ML17929, but I only have one phone number

Normally if you just have one line you use the line 1 jack, I don't know if that would cause your issue or not, I doubt it. If your plugged into the line 1/2 jack, that uses the 4 pin plug and it's showing up on line 2 your jack is wired wrong. Put the cord if you have a 2 pin cord use that just in line one and see if you get dial tone on line one, if not then it's for sure the jack is wired wrong. If that's the case you can either rewire the jack so the line is on the correct pins or you can get a y adaptor with line one and two it will be labeled as such, then plug a 2 pin cord into line two on the adaptor to line one on the phone. So, could your set up be causing the problem, I don't know, but I doubt it as this phone doesn't have a data link. The problem is more apt to be your carrier. Did you have the same issue of slow disconnect with your other phone?

I moved a year ago, about 6 blocks away.  I have the same carrier.  They screwed up the moved but finally everything was ok.  I used this phone for a while is the old house and didn't have this problem.  I had other issues on the previous phone like not being able to dial while on speaker phone (hands free).  It would come back with " your call cannot be completed..."  or  "this number is not in service".  This is why I got the new phone but was only able to get a 2-line.  When I moved is when the disconnect problems started on the new phone. But I sometimes have the problem with the hands free dialing as well.  The phone company has been out several times.  The service rep, who had been a repair person previously, said I needed to have inside wire rewired because they had tried everything.  Problem is, it is an intermittent problem.  When the repair people come out and test the line they say it is ok and won't rewire.  Like I said this has been only on for a year.  They put a monitor on my line a couple of months ago but haven't gotten back to me with how they plan on resolving the problem.  I have been keeping track as well.  The telephone company switch over to voice over internet protocol a couple of years ago about.

If you have a plain old, corded phone plug it in for the appropriate amount of time to determine if it's the phone or the carrier. It sounds like a carrier problem to me. Your statement of when you plug it into the line 1/2 jack on the phone it works on line two instead of one is a bit confusing, that's why I'd like you to try a plain corded phone. 

Correction.  On the back of the phone there is a:  data port, line 2 port, and line 1/L1-L2 port.  If I plug into Line 1/L1-L2l, I don't get a dial tone. If I plug into the data port, line 2 lights up when I pick up the receiver. If I plug into Line 2,  line 1 and line 2 lights up.  When I put the receiver down line 1 lights up  and the screen says "Voice Mail: L1" .  I haven't tried plugging in my POTS phone yet because it means I won't be able to speed dial.  But I will do it now.

In the manual found here it tells you what those ports are for, page 7 for the data port and page 6 for the other two. You are wired incorrectly if you can only get DT on line two.