Typically with one of the two bulbs on a side out, there is not enough load for the flasher to "blink".
So, a new bulb may solve the flash issue.
For the turn switch, try a shot of contact cleaner.
I am not suggesting dirty contacts, but it is a safe way to do a bit of internal cleaning, without disassembly.
CRC QD 11 oz. Contact Cleaner-02130-6 - The Home Depot