I have been in the same situation, written a paper, and wanted to disseminate the findings to a wider audience and hence, presented the work at a conference. I am in applied environmental science, and I feel that this is relatively common in my field.
My approach to this situation, bearing in mind there was to be published conference proceedings was to focus on a very specific element of the new methodological development that I worked on, and presented this at the conference, under a similar - but 'different enough' title than the paper. This parallel journal/conference writing also has to be used to be able to honestly answer the journal question, 'is this work being submitted elsewhere?'. Clearly you need to be able to answer no - its not.
This approach allowed me to go into some very fine detail of the new methods that was of interest to the specialist audience at the conference, that otherwise was far too detailed for the journal readership.
At the end of the conference presentation, I also signposted to the paper that was under review and the journal it was with, and some folk said post-conference that they would look for the journal paper when published.
So, in short, I would say that there has to be differences between the two submissions, but there is no problem in having them linked by common ground and to highlight an upcoming journal paper.
All the best.