With your buget of $350 these are the best options:
Sennheiser 6XX on Drop (Open) - $220
Schiit Fulla - $99
Modmic Uni - $49
Total: $368
(HD 6XX are identical to HD 650, which retail $369. These are considered “higher end” than the options below, which are considered entry level).
Sennheiser HD 599 (Open) - $199
Modmic Uni - $49
Total: $248
Sennheiser HD 569 (Closed) - $169
Modmic Uni - $49
Total: $218
Beyerdynamic DT 990 PRO (Open; 80 ohm) - $179
Modmic Uni - $49
Total: $228
Beyerdynamic DT 990 PRO (Open; 80 ohm) - $142
Modmic Uni - $49
Total: $191
Keep in mind the DT 770 or DT 990 come in 250 ohm options, which requires a Headphone amp, like the Schiit Fulla. So if you get the 80 ohm version, you can use your computers sound card without needing an amp. There are better headphone amps, but that one is inexpensive. And it has a mic input to plug your Modmic Uni into (many do not).
(The DT 770 and DT 990 compete directly with HD 599 and HD 569. The DT 1770 and DT 1990 which are in the $500 range compete with the HD650/HD6xx/HD660S. Sennheiser doesn’t have closed at the mid-range level, only their entry level HD569. They do have a top-end closed HD820 at $2000, but that is overkill).
The Modmic Uni also doesn’t have as good of a mic as the Modmic USB (which has a dual mic system I believe), but you want your Mic on the same DAC so you can do latency free mic monitoring (where you hear yourself talking so you don’t yell). And the Uni is better than any built-in mic on any headset anyway.
I highly recommend open over closed for games. For music it depends on the type. Rock, and any style of music with strings generally are better with open. Any type of music with heavy bass will sound “better” in closed - or maybe “feel” better. I am somewhat weird in that I still like open, but my nephew who is a bass head prefers closed.