Why are they in trouble? What's going to happen to them if they don't have a consensus #1 choice on April 22nd? It's not as if they get penalized or are at risk of being passed. So what's the trouble?
I would be surprised if the other 31 teams in the league have stopped discussing and debating their options and slotting at this point. The Rams, as of a couple of hours ago, had three clear ways they could go with their pick, four if you include Gholston. Some coaches/executives probably like one guy while others like someone else.
Those are discussions that, IMO, probably continue all the way until the pick is made in many instances.
You're right - you're saying they should try to sign someone the day after tomorrow.
And again, as you said, they have four days to explore all of their options. Why limit themselves? Why create a new deadline of two days when you already have one in place of four? Just so you can start negotiating early? Are those extra two days going to make or break contract negotiations with their pick? I doubt it.
I don't see the harm in waiting it out at this point. It gives you time as an organization to come to a final consensus on who you want, and it also gives you enough time to try and force teams wanting to trade up to increase their offer as the deadline nears. And even if they don't begin official negotiations, it likely gives them time to start drafting their initial proposal for their guy since the bar has been set.
Again, go back to my first post. I said the Rams will probably wait to see if they get a great offer as the clock winds down. If it doesn't look like they'll be getting that offer, then maybe they make the pick early. But if the Saints are waiting until Friday night to put their final offer on the table, and the Rams come out Thursday and say, "Alright we're moving on," then that doesn't help them.