I think I understand it now; correct me if I'm wrong. You had a male DIXON take the Y-DNA test. That test only works on the all male DIXON line. With that test you are only looking for matches with other DIXON men; to prove your DIXON line. That Y-DNA test won't tell you anything about your grandmother Ruth DICKERSON.
Whoever took the mtDNA test will get matches with his or her mother, her mother, her mother, etc - all the way back on only the female ancestors. To work with Ruth DICKERSON, you need to trace her daughters and her daughter's daughters, etc down to a living person. The living person could be a man, but his mother's mother's line has to go thru Ruth DICKERSON.
For the Family Finder DNA test: when you open the first Tab of the Family Finder part of your FTDNA personal web page, it will come up showing only your Close cousins (which typically is only a few - two is not unreasonable. There is a little scroll down menu just above the list of your match names - click on that menu window and scroll down to select "Show All Matches" - I'll bet you have at least 50.
You might not recognize any of these names. Each of these matches is probably a cousin of yours from one of your ancestors. How many of your ancestral surnames do you know? 20, 50, 100, more? The other person, each of your FF matches, has a large number too. If they were looking for a DICKERSON, they would skip over your name, too. To find common ancestors with your FF matches, you need to share all of your ancestral surnames with them, and even if you match on DICKERSON, after you've shared, you still have to determine if you share the same DICKERSON as a common ancestor.
It takes some effort to share and agree on a common ancestor, and not all of your FF matches will be willin to share. Heck - 1/3 of my matches refuse to reply to my emails. Some are just starting and don't know much of their ancestry; 6 of mine say one of their parents was adopted; one of my matches was the product of a sperm donation; another was adopted on the black market. But another 40 of my matches were cooperative, and we're still looking for a common ancestor. The common ancestor may be one of our ancestors we don't know about yet - just on the other side of one of our brick walls.
Let me know if the above helps, or not. I'd like for you to get something out of this new test, and I'm happy to help, if I can.