The way I handle lots of games is to prioritize them. My games are listed in order of "Last Move" with the most recent games at the top. Say tomorrow is the 27th January, I work up the list, opening tabs for any 7 day t/o games last moved on the 20th or earlier, 3 day t/o games played on the 24th or earlier, 2 day t/o games played on the 25th or earlier, 1 day t/o games played on the 26th or earlier, etc...
I play through all the games in the open tabs. Once I've done this I at least know I will not have any timeout problems for the next 24 hours.
Then If I still have time left for playing I either cherry pick through the remaining games, or I go into the "next waiting game" loop.
Occasionally my family leave me alone long enough to deplete all the waiting games, but that is very rare indeed.
Although I usually have 100+ games in progress, I seldom make more than about 30-40 moves a day. But using this approach, I find I can comfortably avoid timeout problems.