I'd definitely not do the mulch. Washout and water retention by the mulch is not what you want.
We have a problem with the sump pump drains out right next to the house so we added an extension hose. The problem with that is constantly having to move the hose. This summer we are going to put in a bubbler system. Water is piped away from the house underground far enough so it does not drain right back beside the house. Then there are a series (probably 2) of small catch basins (look to me like plastic buckets with holes). These are connected by PVC piping with holes. This is all installed underground, pipes placed in a trench with large gravel under them. Think a smaller version of a septic system. Going to be a mess to do but I've wanted to attack the lawn, it's kinda lumpy. Large project to say the least.
Don't know if this is going to be an option for you. Need more info on your yard layout.