"As long as your train has a marker, you may not play on your train." is incorrect; the official rule is that if you cannot play any of your tiles and must draw, and cannot play the drawn tile either, then you must put a marker on your train, and your turn is over. This marker makes your train eligible for all players to play a tile on it,
including you. Once
you are able to play a tile, you can remove the marker, which means thereafter only you can add a tile to your train until the next time you must lay down the marker. (In other words, the "penalty" for not being able to play a tile provides more opportunities for the other players to play a tile.)
If you have any tile you can play on an eligible train, then you must play a tile, and you do not have to draw. Exception: if you play a double, you must immediately play another tile on it or draw.
See
Mexican Train Rules and Strategies for the complete rules. Have fun, this game is very entertaining with a good bunch of friends!