fortunately you have some time to plan before than. Make sure you know the exact names and spelling of where you want to go, like "Burg Eltz". It really makes it easier to find these places on the Internet and on maps.
I'm not sure why it is Burg Eltz or the Romantic Road, unless it's just a matter of available time and seeing one wouldn't leave time to see the other. Burg Eltz and the Romantic Road are not that far off the line between Köln and Rothenburg to make them mutually exclusive.
Most of the Romantic Road is actually beyond (south of) Rothenburg on your approach route. There is a small loop, from Würzburg (the north terminus of the Road) to Rothenburg, that would add time, and you might want to consider whether the towns on the Road are worth the extra time it takes from Würzburg to Rothenburg. I did this loop in 2007, stopping in Bad Mergentheim and Weikersheim. I enjoyed these town, particularly Bad Mergentheim and the Deutschesordnung Museum, but I'm not sure the extra time to get to Rothenburg is worth it, and it's the towns, not the Road itself, that you want to see. However, once you get to Rothenburg, some of the towns to the south - Dinkelsbühl, Harburg, and Nördlingen - are worth a visit.
Most of these places are directly accessible by train, and a few, like Dinkelsbühl, by bus (from Nördlingen). You might want to consider if the extra cost of a car is justified.