You need to add a Node layer that is now at ArcGIS a point layer with references added to each line to link the node number. This was built into ARC/INFO so it was really easy. Now you need a third party tool such as Renode Polylines to make this easy such as in ETGeotools. http://www.ian-ko.com/. Just adding points to the ends does not add a single point at the ends, you get two unrelated points. Sigh, this takes a lot of extra steps to cleanup.
When you have properly related point-nodes and polylines you can select all the unsealed polylines and tag the nodes with an 'unsealed end' tag. Now you can select all sealed segments to see if both end nodes have an 'unsealed' tag.
Each road segment will have a field FnodeID and TnodeID. The node points will have a NodeID. Segments that share a node will have the same id. At T junctions the logic is a bit more complex. You will have to tag those as 'unsealed' if all roads are unsealed connecting it. Another useful tag on the nodes is a valence (the count of connectors) to help identify these by running a frequency on the segment end IDs. All of these steps can be done with an SQL query or a join and calculate, don't step through each segment with a cursor!