I've tried (unisex) Depend pull-ups and Tena Pull-ups and whilst both are Ok, they aren't brilliant, perhaps because the padding doesn't come up very far at the front and back. Maybe the padding isn't as wide as it could be either. Having said that, I've had more substantial nappies that have still leaked. Anyway...I wonder whether part of the leaking is to do with the absence of 'leak-guards'. (I assume you know leak guards are the inner leg cuffs that 'stand up' within the nappy and hold the fluid within until it can be absorbed.) I don't think Depends pants have leak guards, I know Tena 'Discreet' Pants don't have them, but I have seen other Tena pants in SuperDrug (they might be called Tena Super) which I believe DO have leak guards. I have never tried them, but I read the packet in a store. I know they would be more absorbent than Tena Discreet, so they might be more absorbent them Depends Pants.
To really get more absorbancy, finding some tape-up fitted nappies would be the ideal, like others have said. So it's Boots Stay Dry, or those shops that sell medical things (or things for caring for the elderly).
I notice you say getting supplies over the web is not an option. I know there are numerous reasons why using the web for these things might not be feasible, but you might be interested to know that the (UK) Tender-Care website allows you to print out an order form and pay via cheque rather than needing a bank card.