How can I best approach journals and academics about paying to re-use content from their papers?
Many journals have a "Request Permissions" feature on their website. Open the paper of interest and look for a button with that label - often it's grouped with other options like downloading citations, setting up alerts, etc. Then follow the instructions given.

Although I've seen this in a lot of the major journals in my field, I've not done an exhaustive survey of how many journals have this option, so your mileage may vary. But it should be a good first place to look. To take an example from your comment, the Bulletin of the American Schools of Oriental Research is on JSTOR, and if you visit the page of an article, there is a "Request Permissions" button right at the bottom, below the abstract. Example here - I hope that the link is stable. If not, visit the latest issue and open any article.