Running is more of a solo activity.
It's not like you can hold hands and have a convo while experiencing the joys of running together... Makes no sense to do it as a couples activity if it isnt already an activity you both enjoy.
Even then, both will experience it individually as you run alongside each other in silence, reflecting on your thoughts, separately... and then what? Discuss your now clear mind, epiphanies or the wonderful sweat you broke out????
Can you tell I'm no runner either?
Sorry, just doesn't compute in the context you describe the need for new activities.
Unless of course you do as Perna suggests and play coach. That's more interactive and makes more sense.
How bout photography?
You can do that with phones, no need to purchase extra fancy equipment. You can pick out locations and pack a picnic for lunch... Then you can create a scrapbook together.
Just an idea, not sure if it was suggested already.