Are either of the monitors you've tried multi-sync?
I'm thinking this is a scaling issue too. The monitor I use allows 1:1 / scaling off but it results in a tiny display in a sea of enormous black borders. It makes my TT look just like Hatari! I mainly use a Matrix CoCo card rather than the Atari video output though as that can output a few of my multi-sync monitors native resolutions, albeit at an eye watering 30Hz refresh.
What's interesting is my screen doesn't have the lines when scaled using the TT output so maybe multi-sysnc monitors where scaling can be turned off can handle the TT better?
I'm using a Compaq TFT8030 which is pretty good for old Atari stuff on the whole.
TT030 10mb/16mb/1Gb, Coco. Falcon 030 Stock 14mb (Lynxman), 1Gb CF. Falcon 030 Stock 14mb (Atari) 170mb IDE. Mega 1 big button (mash up badged Mega 4) 4mb (home cooked RAM upgrade) ICD ADSpeed ST OverScan. Mega 2 big button (4mb), Mega 1 small button (4mb home cooked 72pin SIMM slot), 520 STe (SIPPs), 520 STFM (4mb using a Falcon 4mb board + Blitter).