Heatmap
A heatmap avoids a multi-series line chart's line overlapping problem. One can easily trace each state's trend.
However, using color in the heatmap to encode the data is less effective compared to the line chart's position encoding.
Tile Grid Map
This solution moves each state's row in the original heatmap into a United States tile grid map. This layout can be used to emphasize any geographical pattern in terms of state level trend.
Tile Grid Map Small Multiples
This solution breaks the previous tile grid map into multiple smaller maps, one per each year. This layout can be used to emphasize any geographical pattern in terms of national level trend.