Drew Brees had the most games with 350+ passing yards and 3+ passing touchdowns, with 38 games.

what are the most career games with 350 passing yards and 3 passing tds
Interpreted as:
What are the most career regular-season games by a player with 350 or more passing yards and 3 or more passing tds?
BreesBradyManningRoethlisbergerMarinoRiversRyanFavreFoutsMahomesStaffordManningMontanaLuckRodgersFitzpatrickJurgensenCousinsYoungEsiasonPalmerPrescottWinston3830242019141313131212121211111010999877
NAMEGPYDSTDCMPATTPCTAVGYDS/GTD%INTINT%SCKSCKYRATE
Drew BreesDrew Brees3815,0331431,1421,57672.59.5395.69.1241.548363126.1
Tom BradyTom Brady3011,6231168491,20270.69.7387.49.7141.245278128.5
Peyton ManningPeyton Manning249,614916971,01968.49.4400.68.9242.426156118.3
Ben RoethlisbergerBen Roethlisberger207,8746961788769.68.9393.77.8141.637235116.4
Dan MarinoDan Marino197,7617251886559.99.0408.58.3303.524190102.7
Philip RiversPhilip Rivers145,4234540357869.79.4387.47.8132.21988115.9
Matt RyanMatt Ryan135,0834637452371.59.7391.08.861.130199126.7
Brett FavreBrett Favre134,8124335753267.19.0370.28.191.717135115.6
Dan FoutsDan Fouts135,1304534956661.79.1394.68.0223.9101.5
Patrick MahomesPatrick Mahomes124,8394635449571.59.8403.39.371.41698127.5
Matthew StaffordMatthew Stafford124,7624334754663.68.7396.87.9101.825173110.0
Eli ManningEli Manning124,9114036552969.09.3409.37.6173.224165110.1
Joe MontanaJoe Montana124,7894333650466.79.5399.18.5112.223162116.6
Andrew LuckAndrew Luck114,2154131550362.68.4383.28.2102.020130108.1
Aaron RodgersAaron Rodgers114,3644231046766.49.3396.79.030.628192123.6
Ryan FitzpatrickRyan Fitzpatrick103,8563926940366.79.6385.69.7143.51982115.4
Sonny JurgensenSonny Jurgensen103,9553723538361.410.3395.59.7205.2106.7
Kirk CousinsKirk Cousins93,6142927639070.89.3401.67.441.015107120.2
Steve YoungSteve Young93,5822921832467.311.1398.09.0103.11669121.2
Boomer EsiasonBoomer Esiason93,6923125140162.69.2410.27.7143.521163103.8
Carson PalmerCarson Palmer83,1492925537068.98.5393.67.892.424178111.0
Dak PrescottDak Prescott73,0622422933768.09.1437.47.161.81486112.9
Jameis WinstonJameis Winston72,8632521030868.29.3409.08.1103.21481111.2
Tony RomoTony Romo72,8702421231467.59.1410.07.682.517122111.3
Kurt WarnerKurt Warner72,6722217824473.011.0381.79.083.31044124.9
StatMuse has game-level data for passing touchdowns and passing yards going back to the 1950 season.

See Trending NFL Searches