You have to adjust your People Picker settings for the web application. It appears that your trust is allowing the discovery of user accounts from both domains. If you have disabled the accounts in the old domain you could use:
stsadm -o setproperty -propertyname "HideInactiveProfiles" -propertyvalue "true"
Then create a SearchADCustomFilter to exclude disabled users and groups. If the users are enabled in both you will need to add an additional clause to exclude the other domain entirely:
(|(objectCategory=group)(&(objectCategory=person)(objectClass=user)(!objectClass=inetOrgPerson)(!userAccountControl:1.2.840.113556.1.4.803:=2)))
You can apply all this with PowerShell:
$webapp = Get-SPWebApplication "https://yourwebapp"
$webapp.PeoplePickerSettings #Displays your current values
$webapp.PeoplePickerSettings.SearchADCustomFilter = "(|(objectCategory=group)(&(objectCategory=person)(objectClass=user)(!objectClass=inetOrgPerson)(!userAccountControl:1.2.840.113556.1.4.803:=2))) " #Sets the SearchADCustomFilter value
$webapp.Update() #commits the change
More backgroud and additional info from Ron Grzywacz People Picker Settings
and TechNet Configure People Picker