I would always wait for a reply before shipping as to end up with return shipping costs and complications is not worth the hassle.
Some buyers are always on their PC's and check emails daily others it seems only once a week so delays do happen. Also they may have bid, paid and gone on holiday as happens at this time of year.
Partial refunds are ok if the buyer accepts it providing it covers the cost of replacing that part by the buyer and does not render the item useless.
In cases like this a decent buyer (most are) may be a little annoyed especially after a 10 day listing but would rather you were honest with them. Be pleasant send a reminder say a week on or what ever feels right but do not pester, contact via ask the buyer (keeps all on record).
The partial refund value needs to be offset against relisting again which could be more difficult with a defective part so a little haggling may ensue. Ultimately you want your reputation kept good for future sales even to that person and their friends.
If you do issue a partial refund assuming PayPal then always use the partial refund option tucked away at the bottom of the PayPal transaction details page. Use this and you get PayPal's cut refunded to you, make a totally seperate payment and the actual buyer is treated as a seller and loses funds to PayPal.
In cases where the problem is a seller oversight or damage unnoticed when writing the listing or subsequent to being won (e.g.in storage or even caused whilst packing) I would always readily offer a full refund of their payment should they wish it without question. In other words, my fault, my problem. Of cause this is easier and cheaper to do when you have not shipped the item, shipping before can make the buyer feel intimidated or that it was known to be faulty and you are trying to pull a fast one.
Good immediate and short term handling of a customer will improve long term income.
