I'd head south - Black Canyon of the Gunnison, San Juans, maybe even Mesa Verde.

https://www.nps.gov/blca/index.htm

https://www.durango.org/discover-dur...arks-monuments

I was down south earlier this month and there wasn't any snow - green, green, green. Summer was in full swing. Durango in particular was gorgeous.

Or, keep your plans and dress accordingly - waterproof boots, maybe Yak Trax. That's what we locals do.

My first inclination is stick with your plans for RMNP, especially if you have a reserved campsite that might be snow free. I don't do enough car camping these days to know about finding a site that is "reservable or has a near-certainty of being open". But I suspect that will take some work and luck, especially the latter point these days.
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The high country all over the state has a more or less similar snow depth right now, see map below current as of last night.

Snow at Bear Lake is at 30 inches now and it will really warm up in the next few days. No way will it all melt by mid-June but you can probably slog your way (tiring of course) through it with decent footwear as Mel suggests. Good boots, gaiters, a traction device. Even rent snowshoes if you had to. I've used snowshoes often in June when going up high. Finally many lower elevation trails in RMNP will be largely snow free, though perhaps muddy.



Snotel Snow Depth Map

2nd week of June is not in 5 days, so I'm a little confused on the timing. What day are you currently planning to arrive in RMNP and do you have campground reservations? How many days are you planning to be in the park?

I'm inclined to agree with Nelson - it's not likely you are going to find somewhere that has great hiking conditions with all the snowfall, so stick with what you've planned rather than scramble to find an alternative. There are hiking options in RMNP at lower elevations, as well as some options in nearby national forests. You may even find that some of the tundra hikes are open whereas the forested hikes are snow covered - it's been years but I was there in early June once and some tundra hikes were wide open even though some of the hikes near Bear Lake were covered in snow.

https://www.nps.gov/romo/planyourvis...conditions.htm

That link has some volunteer trail reports but it's a week old. Keep checking for more current reports.

I agree, if you have a campground reservation in RMNP, it might be fine (glorious, even). Plenty of trails will be available. It's going to be pushing 90F in Denver this coming Monday.

The forecast in summer calls for a good chance of rain pretty much every day. That's the way it is in the mountains, there are afternoon thunderstorms but the day is rarely a total washout.

I would research the Tetons as an alternative but you'd want to combine with Yellowstone if you have a full week. The Black Hills area is another option. But really, give it another week and check trail conditions in RMNP and I'm sure they'll be improved from today.