You may run into a problem in the fact that high strength and low dexterity really pushes you toward the slower weapons – but more due to the scaling than the weapon’s requirements. And truthfully, most of the one-handed martial weapons are going to get at least a little bit of benefit from some dexterity.
So for example, the basic longsword has equal scaling for both strength and dexterity. So you’re going to get more damage out of it with 40 of both strength and dex than you would with 65 strength and 15 dex.
Compare that to the black knight sword, which scales more with strength than dexterity. As long as you meet the requirements to wield it, you could neglect dexterity and just focus on strength and still be getting most of the damage out of it.
So now that I’m thinking through it a little more completely, the basic club would actually be a great upgrade choice. It has great strength scaling and has a decently quick moveset, and you’d get to keep your shield. Or, if you want to go off the rails a little bit, you can use the Iron Golem soul to make a Dragon Bone fist weapon.