I'll discuss feeling getting about exercise in a minute but first I wanted to address something. You say you're doing 350 minutes of exercise a week=50minutes/day. Now, there's nothing wrong with that. I'm a former athlete and would definitely be doing at or more than that when I was training. So if you were doing that much because you enjoyed it, great!
However, it doesn't sound like you enjoy it. It sounds like you feel like you HAVE to do that much, which simply isn't the case. Actually, I'd even argue that at some point you're going to have diminishing returns for exercise, especially if you're doing massive amounts of cardio. When it comes to strength training, I wonder if you're even able to lift remotely heavy because your body has to be completely exhausted? So if you're doing that much exercise to lose fat/build muscle, you don't need to be doing it (and it might be conterproductive).
Look into both HIIT and heavy lifting (New Rules of Lifting for Women is AWESOME). It can give you specific routines to do and discusses this vary issue. If you seriously do want to build muscle, lose fat the best thing you can do is give your body rest. Without rest that you're not going to be able to:
a. lift heavy enough to seriously build muscle
b. give your muscles time to repair themselves after you tear them during a workout
c. are at a higher risk for injury.
That doesn't mean that you can't workout often but plan out your workouts so you're at least getting rest days in between heavy lifting (you could do light cardio like swimming or yoga in between but nothing too strenuous). Also, try doing HIIT after heavy lifting and then giving your body rest the next day. Part of your boredom/frustration with exercise might very well be that your body needs a rest.
As to guilt. I get it. Like I said, I'm a former athlete. I'm used to working out A LOT and I do love it. BUT I have injuries, I'm not a young early 20 year old who can pull that off anymore. Instead I need to listen to my body. I still workout often (swim laps 3x a week, heavy lifting 1x per week on average) but I've learned that I can and DO lose weight, build muscle doing that. I don't need to push myself to any extreme to continue.