You have several options in the 300s. I live in east Dallas in the M-Streets area. It is great if you are looking for unique houses with decent yards very close to uptown and downtown. I love it and work in the area. It's close to White Rock Lake, Mockingbird Station and you can also walk to restaurants and bars on Greenville. Very close drive to uptown.
If you are single or looking for more of the urban feel uptown is the place to be. You are mostly looking at townhouses or condos in this price range to be in uptown. The best bars, singles and nightlife is in uptown.
Knox Henderson area is the up and coming artsy, trendy area across from 75 from uptown. It has a little cheaper home price than M Streets but is just south of it. Cochran Heights is a neat neighborhood with similar home styles as the M Streets but just a bit cheaper. The area just north of Henderson is a transitional neighborhood and home prices are increasing.
As suggested by others, Bishop Arts/Oak Cliff is a great new trendy spot in south Dallas. It offers great new restaurants and trendy spots. However, be prepared to be a good drive away from uptown, downtown or greenville. It is across 35 and about 15 minutes south of the city. Great new up and coming neighborhood though.
I think these areas fit you the best. I am in the exact same age range and situation so I can relate. I would like to talk to you further about your search. Call or text me at 817-312-6034 or you can email me at firstname.lastname@example.org. In the meantime check out my website at kyle.wallergrouphomes.com