Installing via Dropbox the way you described it won't work. The installation doesn't just copy files to the computer. It has to interact with Windows to make entries in the registry and for other purposes. It's not a matter of "file signatures," but the installer has to run on the computer that you're installing the software on.
Do what mesquitebean suggested. Get a download and install on the new computer from the download.
Another option, of course, is to get an external optical drive for the new PC. You can get one for around $25. You'll find other uses for it.
