Do not plan to stay in northern San Diego County, like Carlsbad, and drive to Anaheim for days at the Disney Resort. There is basically only one route through there (I-5) through Camp Pendleton and through southern Orange County until you can break out and go west on the 405, north on the 55 or 57, etc. The toll road alternatives aren't convenient at all for Disney/Anaheim so all of the morning rush or go home traffic is going to be going the same ways you want to go. Or, you can hold off and get there later. Disneyland is crowded during the summer even on hot weekdays, so you'd lose a fair amount of time trying to avoid or sitting in traffic. You'd be much better off staying near the resort to minimize travel times, etc. If you want to stay in the greater L.A. area and see more of L.A. than just Disneyland that opens up a lot of possibilities. Carlsbad to San Diego? My impression is the San Diego "rush" is less intense and doesn't last as long as L.A.'s but I don't typically go to San Diego during the work week. My daughter does but not using the I-5 coastal route.
If you aren't planning on being in the L.A. area west side,the Santa Ana and Ontario Airport alternatives should be considered. LAX is busy, crowded, and traffic in the area is generally very heavy, you'll spend more time than you might like just getting to/from the airport. Unfortunately, fares are usually better to LAX and sometimes a whole lot better. Urban traffic is bad in the Bay Area and greater L.A. area mostly because of the volume, but my wife just spent some time in NYC and reported the traffic was horrendous. Gridlock was common and the response to having hung oneself up in an intersection was to honk, not avoid entering intersections you know you won't get through. My impression is that L.A. pedestrians will not shove tourists into traffic. "Nobody walks here" and "Nobody drives there" aren't all that true. Outside of the big urban areas, Ca. traffic seems fine. However, if your idea of "heavy traffic" is a two land divided highway, moving at worst slightly under the speed limit, 5-8 lanes of traffic moving from stop to 50-60 or faster and back to stop and go again, with two lanes or so of truck traffic where you want to merge on and off as well as motorcyclists splitting lanes along the diamond (car pool restricted) lanes, traffic may come as a bit of a challenge to you. Off freeways, traffic is traffic. Cities are congested so if you've driven in New York, Boston or Washington, DC, and survived, L.A. wont seem bad. San Francisco, though is 3 dimensional with toruists wandering everywhere, so it's a bit of a bigger challenge.
Carlsbad is apparently fine. I haven't really been there except passing through for years so don't have any really useful recommendations. My daughters were never interested in Legoland and I think were too old for it. Not sure but have never been there. So don't have any ideas on lodging and restaurants but would expect little problems in finding nice places.
"L.A." and San Diego have an amazing array of attractions, restaurants, communities, foods, etc., and I guess to get answers, we'd need some help as to how old the family is, what you like, what you don't, how adventurous you/they are, etc. I'd suggest looking at the San Diego Zoo and Safari Park night time offerings. In years past, they had added entertainment, etc., in the evenings which was nice as the animals and the tourists are more comfortable after the heat of the day.