I am not a running shoe expert.
I've done a few marathons, done as much as 60mi in a week, but in no way am I a 'runner' nor do I claim explicit in depth knowledge.
Terms for the runner and the tennis player seem, to me, to get blurry. When tennis players talk about "support" in their shoes, they are generally talking about ankle support or their perceived stability of the platform of the shoe to prevent rolling.
Running shoes are very different. If you are getting a "support" shoe, you are talking about your -arch-. Arch support in a running shoe assists those runners who's arch falls or sags when weight is placed upon it. It really does not have to do with how high a person's arch is (ie. I have a high arch, when weight is placed is on the foot, the arch does not fall. "support" is not required).
The first time I went to buy legit tennis shoes I said I have a neutral foot (do not pronate, supinate or need stability for arch support). The sales guy started talking about support shoes. I repeated I had a neutral foot, he kept talking about support shoes. The support of a tennis shoe guy does not equal the support to a runner or person with a running background.
What does this mean?
Do what ndh suggested. Go to a running specialist shop. Not Dicks, not Massive Shoe Store mall guy. Running store specialist salesperson will have you take off the shoe, stand up, sit down, stand on one let, may even put you on a treadmill to watch your stride. THEN get the shoe they think is right.
I personally can't stand tennis shoes. They all feel totally over-built and heavy compared to what I use in other activities. I liked the NB 851 (discontinued). Next pair I get will probably be the Asics Speed Solution or the new Speed Slam shoes...based almost solely (see what I did there..that is funny) on their weight.