One more thought. You don’t need 375 signatures. You need 800-1000 signatures. Turn in 500 signatures, and your opponent will work night and day to disqualify 126 of them.
It’s far easier to remove an opponent from the start than it is to actually run a primary.
If your district allows you to buy onto the ballot (fee vs. petition), pay the fee. George Washington never gets rejected by the filing office. In addition, you want to spend every second getting your message out, and spending time explaining the petition shouldn’t be part of your message.
I helped gather signatures for a local recall election and city officials disqualified hundreds of valid signatures. They disqualified whole sheets of petitions because an initial wasn’t in the right place. Don’t assume because a signature is valid, it can’t be discounted. You’ll need enough signatures to make the effort to disqualify not worth the time or energy of your opponent. In other words, you’ll need more than twice the minimum number.