Shimano new shifters needed to replace 105 10speeds

The ST-R4600 is a Tiagra model of shifter that is 10 speed but follows the the engineering of all Shimano 7, 8, 9, and 10 speed road shifter/derailleur's having a cable actuation ratio of 1.7:1 (derailleur) and the shifter pulls the necessary amount of cable. With the release of Tiagra 4700, the shift actuation changed to be like 11 speed road despite remaining a 10 speed group set. It's generally felt that Tiagra 4700 components are only compatible with other 4700 parts. A newer model of shifter for 10 speeds is ST-R460. It may be easier to acquire this 10 speed shifter as the 4600 is out of production and used or NOS is the likeliest options to get one of those. You only need the right shifter as left shifters are the same

Any 10 speed Shimano and SRAM road bike shifter should be compatible, except for the Tiagra 4700 which has a different cable pull ratio. Edit: As Pisco points out in the comments, 10 speed GRX (and potentially future 10 speed groupsets) uses this new pull ratio as well.

You could try to find used Tiagra 4600 or Shimano 105 5700 shifters.

You could go for the Tiagra 4700 shifters and also replace the rear derailleur for a Tiagra 4700.

If your cassette, derailleurs and chain are worn down you could replace all of it and upgrade the whole drivetrain to Shimano 105 7000.