A little of the background still shows around my entire selection - what should I do?

If the selection is not active, load the selection again by (1) press CTRL/COMMAND and click on the layer thumbnail or (2) with the layer selected, go to the select menu > load selection. Then go to the select menu > modify > contract. Try a small radius amount, like 10. Then create a new layer from the new selection and hide/reveal the original layer for comparison.