These aren’t bins of glowing green ooze like in the cartoons. Most waste products are solid and are stored in heavy duty, shielded bins. Kyle Hill on YouTube created a really good video that demonstrated walking through a storage facility and literally hugging one of the bins. His rad meter was perfectly normal the whole time. He also examines other risks as well as the risks of other methods of power generation.
Video for reference: https://www.youtube.com/watch?v=4aUODXeAM-k
Here’s the one where he hugs the waste casks:
Look for local craft or art shows. This can help you find local crafters that make some common items like socks, soaps, etc. Hand-knitted socks are the best.
Other than that, for more commercial things, watch labeling (as you already are), go to farmer’s markets, choose local grocer cooperatives (they’re more likely to carry locally produced goods), etc.