I had this same problem, and at times I still do...
The best bet is to take a trip to Thousand Needles (if you are not on a pvp server), Hillsbrad Foothills, or a few places in Duskwood to get some iron and heavy stone, build Iron Grenades and sell them to a vendor, this will help level up your skill, and one grenade will sell for 5s. You will always make 2-4 Grenades, so that is 10-20s. I have generally not had much luck with the AH and my engineered stuff, so I use this quite a bit to get some money.
Another good idea would be to make an alt with enchanting. Do a few runs through the Dead Mines and send all your greens to your alt. Disenchant all the stuff you get, and sell it all on the AH...
I would agree that you would regret it if you drop engineering, especially with some of the stuff that engineering offers: iron greanades that give a 3 second stun; a remote to control mechanical enemies, and other things...