Sure, you can run 93 octane, you can also run kerosene and lighter fluid for that matter LOL
On a 250, octane isn't as critical as a 500 so 93 should be ok in most cases.
I've seen better performance and more consistent jetting with 100LL (avgas) though.
If you've done some performance upgrades like porting, pipe, and haven't had the head cut for optimum geometry, you can still run the risk of detonation even with a 250.
I had a mostly stock 250 (FMF pipe and aftermarket reeds) and when I ran 87 it destroyed the piston after some hard runs in the dunes.
Wish someone had told me the same thing beforehand, that was a long walk.
So I guess the risk is yours to take, I won't tell you not to but it's difficult to deny the convenience of running pump gas.

As for your other problem, you need to find out whether it's an instantaneous issue or something that takes a few seconds to develop.
Go easy with it and see if you can get up into 6th gear with light throttle.
If no problems arise, slowly give it more throttle while you hold the brake a bit, maintaining speed.
If it starts to bog down with throttle position, and gets better when you let off the throttle, you have a jetting issue.
If it starts to bog down while simply giving the engine a decent load and doesn't get much better when you let off slightly, it's a fuel flow issue.
Use caution with both, as they could be running dangerously close to melting a piston from a lean condition.

For the jetting, you'll need to find out whether it's rich or lean and adjust accordingly.
Engines tend to need slightly different jetting requirements after a top overhaul, and it's also starting to cool off which changes jetting as well.
Your jetting may need some attention, and it's easier said than done if it's your first tuning.

For fuel flow issues, close the fuel petcock and remove the fuel line from the carb.
Hold a bottle to the line and open the petcock...it should flow freely out of the tank.
If it doesn't, clean the tank and/or screen inside, or see if there's a problem with the petcock of fuel line.