my 2006 mazda 3 is making a draging noise as i am coasting down the road

joshmazda3-hi there I went to 4 places j and L transmission in portland oregon-they said transmission , went to paradise auto in tigard oregon - been to them before they told me carrier and pinion was bad and thats the differential gears, went to dealer in beaverton oregon they wanted to take my money and tell me i needed a new transmission $4,200 i said no way then last AMCCO Transmission in tigard oregon they told me transmission differential problem lower end $1800 high end to $3,000 I could not get a loan from them that would pay it so right now its siting a my apt why i save for it. o by the way as i was fixing my wifes break pads on her car a man in the apts asked if i needed help with the breaks i said yes then we talked about my mazda 3 i told him what the first person said carrier and pinion he said that it is the differential gears. he is a acura mechanic who has been working on cars for 30years . it is differential that needs work