Playwright accepts a proxy at launch: server with http://IP:12345, plus username and password. For SOCKS5, server points at socks5://IP:4444.
One context per address
Each browser context should have its own address. Two contexts on the same address pool their reputation; one context switching address halfway merges the cookies.
If authentication fails, the error shows up as a page failure and not as a proxy error. Check the username and password before touching the rest.