So you genuinely believe that if Everton play the waiting game Southampton will collapse in a panic and do a cut price deal? Personally I don't think making a bid soon after is particularly desperate if you want to get a deal done. Going above what the buyer considers the genuine value is desperate.
There will probably be a delay as you say due to their being other irons in the fire. May well be a couple of other bids on the go so see how it all pans out before escalating any.
I don't think Southampton will worry too much about replacing a squad player. For their needs there are far more options going into a Championship season and they likely already have a list as long as their arm of sub £5M players.