The alternate routes suggested are good choices, but I think you understand that all that traffic that WAS on the 405 will now be on the 5 (and other routes) so there really isn't a great way around it. You really only have two choices. Plan for a longer than usual trip, or take the train.
I've been on Amtrak several times and it really isn't bad at all. I'd like to dispell some of your stated fears.
1) You don't check your bags. You bring your bags with you to the train and they are placed in a baggage area. When you leave the train, you go get your bags and take them with you.
2) The train isn't like taking a public bus. You have plenty of opportunity to get up and move around and there are several "ammenities" such as: a) They have a dining car when you can have an actual sit-down meal, with a waiter, just like a decent restaurant. b) They have a viewing car (the Coast Starlight to Union Station at least) where you can sit on the second floor and enjoy the view while you visit with others, play cards, etc. c) They have a "lounge" car where you can get a variety of drinks, snacks or other food items. It is sort of like a miniature 7-eleven. There is a good variety, but bring cash.
3) You will have assigned seating that is based on which stop you will leave the train. That means you won't see a lot of people getting on and off the train at every stop.
4) I have not looked this up, but your route would likely require you to board the Coast Starlight to Union Station. I think you can stay on that train into Anaheim where you would get off in the parking lot of Angel Stadium. You should be able to get a shuttle from there to your Hotel near DLR. The Anaheim station is relatively small and very clean.
5) You really can sit back and relax and start your vacation as soon as you board the train. Read a book, go over your travel plans, play some games, have a drink, watch the countryside role by or even take a nap.
There are a few drawbacks. You will travel through some very interesting areas in and around Los Angeles. But you will be safely inside the train looking out so in some respects it gives you a chance to see some things you might not otherwise ever see. It isn't really *bad* but it probably isn't what you are used to either. It will take longer than a car and there isn't much flexibility in the schedule.
If you are interested, you can decide to leave the DLR area earlier on your return trip and plan some additional time around Union Station to visit Chinatown and Olvera street. You can take the metrolink, which runs on a more frequent schedule and is more of a commuter train, to Union Station and board your train to Santa Barbara when it arrives. Again, that would depend on the schedule.