Hi, Genobambino! (Love the handle!)
You absolutely do need to bring strength training into your program, and yes, weights will help you break through your plateau. When you strength train effectively, you build lean muscle mass that is metabolically active, so you'll burn your fuel (the calories and other substrates in the food you eat) far more effectively. Plus, the stronger you get in the muscle department, the more force and intensity you'll be able to bring to your aerobic / cardiovascular work.
If you feel guilty after a weight workout, it could be that you could be lifting a lot heavier and harder than you did IN the workout! Don't underestimate how strong you already are, and how strong you can become! And don't hesitate to do your patriotic duty and buy all of the Cathe strengthening tapes available!
I say this to a lot of people trying to "lose weight", but I never stop believing it: go real easy on using the scale as a determinator of body composition improvement. If you lose a pound of unnecessary storage fat but at the same time GAIN a pound of much-needed lean muscle tissue, you'll have done yourself a double good, but the scale is only going to say you "didn't lose weight". Go by how you feel, how your clothes fit, and most importantly how much you can do and improve in your workout execution.
Good luck!
Annette