I have a power-ranked Mario in my region, who I have the benefit of living close to (we go to the same school), so I get to play him quite often.
Play with patience. Mario's only real advantages on us are if he can get in early he can rack up a bit of damage on us quickly, and I guess if we recover recklessly he can cape us then FLUDD us. If he's actively going for the low-percent combos (usually with a Fireball approach or similar), that's a high risk-reward situation for him, as we completely outclass him at close range otherwise. All I can say is be smart, and try to be safe - learn to recognise how they use Fireballs to approach and the appropriate punish/retreat.
After this, like you said, they will run away in desperate fear of not getting grabbed. Take it slow - walking shield, ftilt clashes and occasionally jumping tend to Fireballs for the most part. Don't lose your cool and make a mistake. When you get close enough, do your usual stuff - RAR bair and all that. My Mario tends to pick Final Destination against me so he has the most room to run away (despite how odd FD against a Dedede sounds). If you do manage to get a grab here, that's an easy 50% or so. Platformed stages do help against Mario's camping.
Actually, like all of Dedede's matchups can be described with 'patience.' Not sure if I helped all that much.
Also, I believe we gimp him fairly well. They'll usually use Fireballs to ward off edgeguarders, so try to come in from above and fall on him with a bair. They can't afford to recover really low to avoid this.