If you already lost a chance at a job interview over the issue, why even bother keeping a phone you don't trust any longer? Get a corded phone and use that in place of the cordless for a while and see if the problem goes away. They cost about the same as a delivered pizza. If no more problem, save up for a better cordless phone. It kind of sounds like the phone base-station is taking the line off-hook and leaving it off. The phone company only beeps the line for a short time before making it go silent. When you unplugged the phone base, I figure it is hanging up the line, so you get a dial tone again when you check right away.
A story from my childhood in the late 1970's. That was a time of great change with phone service when the government deregulated the industry and broke up the Bell system. My parents were still of the mindset that the phone company owns all the phones and wiring in the house (was recently changed at that time) and so the phone company would put them in jail or something if they ran a second phone on the line. My mother wanted a phone by her bed because all of my 5 siblings were much older than me and doing the drunken young adult thing. My father was also a cop working the night shift at that time. Cordless phones then cost in the hundreds of dollars. She came home with one after work one day, a "back of the truck" deal. I guess the black market deal was too much to resist.
That phone had problems from day one. It was in the days before basic digital security codes were used. As long as it was on the handset was on the base, no problem. Once my mother took the phone upstairs to her bedroom, the base would come alive. It would click on-hook, off-hook regularly. It would start dialing numbers. Other weird stuff. I could not convince my mother that this was some kind of problem. She was never present with the base to see it. She also did not understand when this was going on that it would prevent other people from calling us since the base is tying up the line. She would not give up her "good" phone and have the money spent on it go to waste. This went on for about 3-4 years before she finally witnessed the clicking base in person one day. We replaced it with a rummage sale cordless phone that worked perfectly when I replaced the battery pack.