To pull this off, Verin would have had to devote much time to establishing a "legend". She would have had to interview Nynaeve at length, and Flinn, and the sisters who were healed, planting firmly enough the notion of her deepening interest in those matters.

She would have had to find something credible to justify going through the experience herself, to meet the barrage of opposition the very idea would have raised among Aes Sedai. Stilling a sister without an edict from the Amyrlin condemning her to the punishment is likely a stilling offense, anyway.

That wouldn't have been that easy in Verin's circumstances (in hiding in the inner circle of TDR, on the eve of TG), with Cadsuane there too. Most likely Cadsuane would have told her to get a grip and take some time to rethink her priorities. A book on healing stilling, on the eve of Tarmon Gai'don? Come on Verin, can't that wait, don't we have better things to focus on right now?

She had already opened her game too much with Cadsuane, who knew all too well by then that Verin wasn't the absent-minded Brown she pretended to be, and who made good use of that fact herself even though I highly suspect Cadsuane made use of Verin in part to keep her closely under watch. She would have wondered about Verin's real motive all along (and not likely to puzzle it out, so go figure what she would have imagined).

In any case, I think we can agree Verin couldn't just have woken up one morning with this notion, ask around and get it done by noon. It would have taken her several days or more, with no certainty she would pull it off and manage to get sisters to do it.

Then, remember Mesaana's methods with her pawns. She has her tricks to make sure they don't omit any information and that they don't lie. When she's in the mood for it, Mesaana can ask you about any little trivial detail, and you never know what she already knows. Verin couldn't be sure she was the only spy for the Forsaken near Rand, she knew for fact Elza too was BA. She would have been in her neat pickle if Mesaana asked her what the heck she was playing at with her obsession about Healing severing and her project to go through it herself...

Finally, Verin just didn't want to pass her information to anyone near Rand. She even deflected Rand's attention from Caemlyn/the BT when Logain tried to convince him to act. She said Taim would have to wait and embarked on a tirade about TG coming any day now.

It's not a matter of trust. The evidence is that she trusted Cadsuane well enough, and Rand of course, and no doubt Nynaeve as well. They're just not the people she decided were in a position to act. Verin has always been fairly careful in how she dealt with the entourage a ta'veren has gathered around him. Verin wanted Rand focussed and Cadsuane and Nynaeve at his side. She could have let him the big book of the Shadow's plans for the Enlightement of the Dragon Reborn if she judged it the best course of action. She didn't. It's Egwene she picked as her main recipient. She must have had her reasons.