Any way to use the Find feature in the actual formula?

You can probably use indirect() and then just refer to a cell containing the sheet name?
something like...
=SUMif(INDIRECT(B13&"!E3:E200"),INDIRECT(B13&"!$J6"),INDIRECT(B13&"!F3:F200"))
where B13 contains the sheet name

If that doesnt work for you, ii would suggest you upload a sample work book, (all sensitive data removed), showing what data you are working with, a few examples of what your expected outcome is, and how you arrived at that
To attach a file to your post,
click advanced (next to quick post),
scroll down until you see "manage file",
click that and select "add files" (top right corner).
click "select files" find your file, click "open" click "upload" click 'done" bottom right. click "submit reply"

Once the upload is completed the file name will appear below the input boxes in this window.
You can then close the window to return to the new post screen.