You do need to use the Merge tool. The Network Dataset will not Planarize the lines for you. To Planarize the lines with a geoprocessing tool you can use the tool that Devdatta Tengshe points out in his response to this post:
"The Feature To Line (Data Management) geoprocessing tool can be used for 'cleaning up' the data.
The help file mentions:
Where input lines or polygon boundaries touch, cross, or overlap each other at locations other than their start and end vertices, they will be split at those intersections; each of the split lines will become an output line feature. If an input line or polygon boundary is not intersected by another feature, its entire shape will still be written out as a line feature.
In Addition, remember to set the Preserve attributes option to true, in order to preserve the attributes in the output file."
Also the comments by ccn and fmark point out that:
"This is the most efficient option, and you can go a step further when you set an appropriate cluster_tolerance to 'correct' minor undershoots and overshoots in data, too (but be wary that the tolerance value doesn't simplify the data beyond useful spatial accuracy)" - ccn
"Its worth adding that if you want to do the cluster combining as a separate step you can use the Integrate tool." - fmark
If you have ArcMap 10.3 you should also look at the Conflation Toolset in the Editing toolbox. The Generate Edgematch Links and Edgematch Features tools may be of particular use. The other Editing toolbox tools can also be useful if you don't want to use a geodatabase topology.