Five to seven minutes of intermittent blowing doesn't sound very long. It sounds as if your lip is being given too hard a job to do. If you are straining to put pressure on the reed something is wrong.
As a beginner a soft reed is a must until musculature builds. What mp and reed are you using? The popular Yamaha 4C with a 1.5 reed or thereabouts is the usual starting place. Be aware that reed strength varies, will be easier to play than others even if marked the same strength.
Only go for harder reeds when the ones you are using are too soft to get the high notes. If the reed is slightly too hard move it back from the tip of the mp a mm or so, instead of being level with the mp tip. Don't have your ligature too close to the start of the ramp on the reed - a bit further back allows it to do its stuff more easily.
The thumb sucking baby embouchure has a lot going for it. A nice relaxed embouchure is the thing to aim for. Try the most relaxed you can while still getting a sound.
Using a relaxed embouchure lowers the pitch of the note, but you bring it back up by pushing the mp on further and between the jigs and the reels you get increased volume and tone benefits.
It will all start to click into place soon. But a good teacher is worth looking out for, even if only to get you started on the right path.
Maybe I should say 'a right path' - there are some choices - we are all different, and so are our instruments, even if the same make and model...