I own an HE-500 and Sine (original) on the planar front as well decent number of dynamic can's (T1, HD650, X00, 325e get the most use). With my cans and experience here are MY generalizations, this won't apply to all but you might be helpful:
1) Bass - Dynamic can have more quantity but planars generally have less rolloff in the lowest frequencies.
2) Mids - This varies greatly for me on dynamic cans, generally pretty linear in planar technologies.
3) Treble - the HE500/Sine don't have the most linear treble and my dynamic mainstays are also all over the board so I can't offer much here as far as generalizations go. Seem's to be more dependent on manufacturer than driver technology.
Soundstage Size: Better/wider on my dynamics but both of my planars have good depth, just a little intimate/narrow.
Imaging: Varies a bit with dynamic... both my planars image quite well.
$350 is a good budget for a solid set of cans... my thoughts on a couple of MD options in that range:
1) The K7XX is reputed to have a great soundstage with a fairly linear sound if not somewhat unnatural. This (well it's predecessor the K701/2) is the only one of the original "Big Three" from Senn/Beyer/AKG that I've not owned so I can't speak to details. Should be a solid choice if you like a big soundstage and will be doing a lot of gaming. Moderately hard to drive due to it's relatively low efficiency. Some sort of amp would be recommended but is not necessarily "required".
2) HD6xx - this is basically an HD650 with new colors, cables, and a much lower price. The HD650 still holds the record for longest continuous listening session in my collection. This was my first +$300 headphone purchase and it still gets head time over a decade later. Really good choice if you like acoustic music and vocals as it sounds more natural than what little I've heard the AKG K series. The soundstage is deep and fairly tall but the instruments/singers tend to be packed a little closer to each other than some of my other 'phones (aka 3 blobs/groupings). I generally don't recommend this with a basic motherboard or low end HP output. Needs some sort of amping or higher voltage swing output.
3) HE4xx - seems to be a good deal based on other 4 series headphones. When I picked up the 500 it was a choice between the original 400 and the 6. While I wanted the HE6, I did not yet have an amp that could handle that beast nor did I want to part with the funds. The 400 was more bass oriented and a bit mid recessed IMO. The 500 is still as close to perfect as I've yet found as far as my preferred sound signature (not perfect, hence the stupid number of HP purchases I've made since).
If you've got no current amp/dac and only 350 to spend then I would go with the above options and spend the rest on some amping you may not appreciate the difference a DAC makes yet depending on your source.
If you like a bit of bass (the 40/40x are pretty neutral) then the Fostex X00's are truly wonderful but definitely not neutral. If you like things a tad more neutral then the new Hifiman Sundara looks like a solid choice and addresses some of the prior build concerns. My HE500's are several years old and I've had no issues but I'm also super protective of my gear.