Common problem with the 5600 shifters.
The 5700/6700 shifters also had this problem, plus the shifting was worse due to the cables running under the bar tape.
I’d probably keep an eye on ebay and just get a second hand set, otherwise it’s gonna cost you over £300 for new 105. New Tiagra could be an option though.
Another option would be find some Campagnolo 10 speed Ultrashift levers as these are compatible with Shimano using a Jtech Shiftmate. Plus they rarely break (unlike Shimano), and if they do you can just strip them down and fix them easily.