If there's snow or heavy runoff, Murphy Creek can be a little ugly, and there are going to be a few spots where water crossings are unavoidable. The valley up behind there along a big portion of the May Lake Trail is heavily shaded, so it tends to hold snow a little longer. With snow, lots of bridging and a little route finding at times. Without, lots of standing water and mosquitoes. Otherwise, beautiful area that's worth seeing all the way up to May Lake. However, you also have to remember that May Lake itself sits at close to 10,000 ft, so with all the snow we've had, with more to come, you're very likely going to be hiking/camping in the snow. In fact, weather dependent, May Lake may actually still be at least partially frozen over.
But, even if you're fine with weather and conditions, the single biggest issue you're having is that any time around the 4th of July holiday is going to be more heavily impacted than other times. You could definitely try for a walk up permit, but the best trick is to just avoid the park in the first place and save yourself from the madness and stupidity. You go to the full trailheads report:
https://www.nps.gov/yose/planyourvisit/ ... lheads.pdf, and you'll see that, even with moving your time out just a few days, your trailhead options open up considerably. Murphy Creek > May Lake > Snow Creek > Valley is a nice route, but it also isn't necessarily the best route, perhaps even more so for a first timer. A good alternative is Porcupine Creek > Snow Creek > Valley...keeps you out of the snow for the most part, more bite sized for a beginner, still a beautiful little route with some stunning views and great campsites. You'll just want to make sure that camping restrictions because of bear activity aren't in-force for the Snow Creek area.