Gosh -- on my selling ID I always -- every time -- take my sold and packaged items to a local post office the same day I print my mailing/postage label through eBay and/or PayPal. If I get a confirmation that the buyer has paid after the time of day I know my local post office(s) have their last pickup , what I do is wait until the next business day to print my label and take the item to the post office. And then, I'd say about 90% of the time, I send a message to my buyer saying "I just dropped your package at the local post office drop box." Buyers are able to click on that computer-generated tracking number and see whether I really took the package to the post office that same day or whether I waited, as you say, "a week or more."
There is a way, though, for you to make sure the tracking number for your sold item does not get entered into eBay's system. Pack up your package or envelope. Either write the recipient's name and address onto the package/envelope or type and print a label that has only the recipient's name and address.
Take that package/envelope to a post office. Stand in line. Go to the counter and pay for the postage and a tracking number. Your printed receipt will also show the tracking number.
Go back to your home and log back onto your computer. Go to eBay and open your "Sold" items. On each of those "Sold" entries, you will find a place where you can enter the tracking number -- so, you will want to type in that tracking number that shows on your receipt from the post office. Be careful and make sure you type in that long string of numbers accurately.
A lot of sellers don't really have time to do all that, and we still manage to keep our integrity.
