There are a couple things that spring to mind:
- Check that you're playing correctly.
If you're straining yourself unnecessarily, then you will become tired much more quickly and will therefore play for less time. The comment that makes me bring this up is your description of how your lips / jaw get numb. To me that indicates that you're clenching with your jaw, which isn't good. As for the lips, I've found that with playing any instrument for a length of time, your lips just get tired and cannot hold the embouchure; perhaps sore feeling but not "numb".
I would check with either an oboist or an oboe / woodwind instructor and have them watch how you are playing. Describe what you feel and see their response.
- Play more often.
This one is just one of those age-old axioms. If you want the chops, you gotta play. Simple as that. Granted, if you can only play 20 minutes at a time, it is useless to try and force you to play longer - you'll only develop bad technique that way. Instead, I would recommend doing what many players do: having at least two daily practice sessions.
The first "practice" session is really just a warmup where you focus on fundamentals: long tones, articulations, scales, range, etc. You do that in the morning at a convenient time. The second session is later in the day, in the afternoon sometime. Now that you're already warmed up, after a couple scales you're ready to do hardcore technique / excerpt work.
The last "warm down" session is optional - some people / performers swear by it and will nay-say anyone who says differently. I've known both types. A trombonist friend of mine warms down for about 15min. after every gig / rehearsal regardless of how long he's been playing. I myself have never needed a "warm down", so it's one of those things.
I don't know when your exam is, but remember that when it comes to instrument pedagogy, you can't cram. You either have the technique or you don't, and with oboe certainly, there is no faking.
Just do your best preparation work possible, practice and play daily, and when it comes time, don't warm up too much before you play - it'll kill your chops and you'll sound bad.