There are a few ways to do it, none of them fun. The first question you need to answer is "Is this a public sidewalk?"
this is the system I use, just cut your width to your post size, predrill everything before going and it goes fast, you drill your holes (did not show it but 3"bolts go all the way thru with washers and locknuts) for the 3" thru bolts a little larger for leveling purposes
If cost wasn't an issue, I'd properly cement some flanges into the ground, set the posts inside the flange and bolt the posts fast. Using your way, if something happens to the sign and/or post, you have a real mess. This was it's like a break-away point if enough force is used and you don't disturb the underground part.
Cost is the issue. You how sometimes you should just walk away? Well, I agreed do this job a LONG time ago, and they were very upfront about their budget. I'm the moron who keep coming up with more expensive ideas, knowing full-well that I'm only going to get a certain amount.
But, on the other hand, this organization referred me to someone else who spent beaucoup bucks this year, so we are happy to do the best we can. Anchoring the posts will work just fine, its not a big sign.
In the future I will remember to budget for a more solid footing.
Thank you for your suggestion.