Julie, I noticed some tasks under the 'Advanced' tab under Information show 'As Soon as Possible' for the constraint type while in other tasks the field is not enabled. Could this be an issue?

Have you customized the view with formatting? You may have luck re-setting the view to default - but you will lose any customization. Display a view other than the Gantt chart (assuming that is the one that is not showing correct information). Click the Task Ribbon and click on the drop down underneath the Gantt Chart button (far left). Choose Reset to Default.

Regarding the Constraint type - are you using Manually scheduled tasks? You cannot change the constraint type for manually scheduled tasks.