If I recall the news reports, Ian was moving about 10mph - I guess it’s debatable how much of a difference it makes to add 10mph when we’re talking over 100mph. Also, winds are fastest near the center (except for the eye) so it goes without saying (but I did) how close to the center makes a difference too.
The movement speed of storms isn’t so germane to the windspeed, but rather to how long it will stay somewhere and rain on it/blow at it. 10 mph is on the slow side for a hurricane, since the average is 15-20 mph.
the movement of the storm is important. Not only is the right side faster, the left side is slower. The amount of rain is also effected, which is usually noticeable. Source: been there. 
And the force of wind goes with the square of velocity, as does the energy of any objects carried by the wind. A 110 mph wind is ~50% more damaging than a 90 mph one.
Actually most of what I’ve read lately is that wind speed is not as great a factor as is the SIZE of the wind field. Large, slow(er), spread-out hurricanes produce greater surge than tight, compact, fast(er) hurricanes. It’s more about how much time the wind has to affect water.
But yes, the surge is contained to the right/bottom side of a storm moving in any way north or east. When a hurricane does what Ian did and crosses the Florida peninsula traveling west-to-east, the storm surge for the east coast on exit is actually on the top/left of the storm. Ian was particularly bad in this aspect because the oblique angle of approach meant both the right side and bottom of the storm had plenty of coast to impact on approach as both a fairly large and very strong hurricane.
And yes, rainfall has a lot to do with it. Rivers are trying to move water towards the sea, when the river is high due to rain and the surge is upriver, the river “backs up” and floods worse.
Several factors play into the size and location of the storm surge. The water depth off the coast is a factor-shallower water further out is bad. When Katrina hit Louisiana and Mississippi one reason it was so bad was that it had crossed over a submerged warm-core eddy in the Gulf 24 hours before landfall. So it got a charge of energy (went to Cat 5 for a brief time) just as it was coming onto the shelf. So more wind energy and shallower water built up the storm surge beyond what would have been expected from a storm of the strength that hit the coast. Ian crossed the loop current, but did that in deep water. The Hurricane Center is very good predicting the location of landfall, strength is OK, size of the storm surge not so much. At least not until it is way to late to do anything.