Based on EuYu, it appears there is no such graph. I originally deleted my answer after that. But, now I put it back so the code is visible. This code will find examples of graphs that satisfy all the conditions EXCEPT the unique Hamilton cycle.
Using Sage, including the nauty generator of graphs, we can generate all graphs with 3 * ord / 2 edges easily. Of course, we only need check graphs with even order as well. So, this should go quick for graphs of small order. But, even at order 12 it might take a very long time. There are over 10 billion graphs of order 12, not sure how many have 18 edges exactly.
# ord must be even
ord = 12
num_edges = 3 * (ord/2)
# This is the string that gets put in the nauty generator to tell it which
# graphs to generate
# E.g., "12 18:18 -c" says generate order 12 graphs with 18 edges that are connected.
nauty_string = str(ord) + " " + str(num_edges) + ":" + str(num_edges) + " -c"
for g in graphs.nauty_geng(nauty_string):
# check min degree is 3. Since we have the constraint on edges above
# this should guarantee the max degree is 3 as well.
deg = g.degree()
if deg == 3:
vert_conn = g.vertex_connectivity()
if vert_conn >=2:
if vert_conn <=3:
print "Finished with", ord