What I'm looking for is a tool where the entire purpose is to draw a graph and have it return the graph6 code (ultimately so that I can easily input it into Sage).
On House of Graphs, you can draw a graph and it will search for the graph in the database. But if you search for a graph which is not in the database, you get a message like this and an empty list.
Your current result list contains all graphs that satisfy the following criteria:
Graph with canonical form 'H???Gva'
The problem is that if the graph is "too big", you get a message like this, where the "..." indicates that we are not seeing the entire graph6 code.
Your current result list contains all graphs that satisfy the following criteria:
Graph with canonical form 'S????????????????????????????F...'
Note: There are tools such as this which can translate an adjacency matrix into graph6 code.