Volume is key. To some people, "Quite much" may mean just enough to make your pants noticeably wet. But to others that might mean a full bladder to others. Take a look at Northshore's website. They have their own pads, and some from other companies. But most importantly, they show absorbency for each product (in ounces). Absorbency ratings tend to be theoretical maximums. You would probably never use the maximum, because it implies wetting the whole pad evenly, which is unlikely in real use. But they do give you one means to compare products. So if a product is rated at 4 ounces, and that's not enough, you can try something with more capacity.
I've found that while people here can be somewhat helpful in directing you to possible solutions, you really need to test some products to see what best meets your needs. And you need to figure out how long you need to wear an incontinence product before you are able to change it. I stress the word "need" rather than what would be your dream. That is because you can certainly find something that will hold an ocean of pee. But even the most discreet products increase in bulk as they get wet. The highest capacity ones will get quite bulky. I can't imagine doing gymnastics in something like that. So you may want something that will last just long enough until you can get to a bathroom and change the pad (or pull up). Changing may still be awkward and less than ideal at times. But you may find that is better than a very bulky pull-up or diaper.
I like Always Discreet Maximum pull-ups, even though I am a guy. They don't make a men's version, but other men have recommended them. They fit close to the body, and are as discreet as anything I've tried (some products I tried seem to hang down in the crotch). They are also very comfortable (some, like regular Depends have the feel of paper towels). The Always Discreet Boutique are even softer, but a bit tighter, which makes the regular ones a better fit for me. I have overactive bladder, so when I pee, it's generally less than 3-4 ounces. As long as I don't let it all go at once, the Always Discreet's will usually more than one wetting (though 3 would be pushing it).
I've worn the Always Discreet pull-ups while running. I haven't had any chafing issues, even after 3-4 miles with them being somewhat full (at most 3-6 ounces).