You've summarized your options well.
Booking.com has the Park Plaza Lodge Hotel (an older 2-star that gets good ratings) for $120 per night PER ROOM. They have 2 room types: a 'queen room' with one queen bed for you and your husband and a 'standard double room' which has TWO full-size beds for your sons. The parking and wi-fi are free so you'd be able to have 2 separate rooms for you and your sons and parking included for about $240 per night. The hotel has a small pool.
Here are reviews for the hotel (link and scroll down for the reviews):
The property is a 6-minute walk from the Grove (http://www.thegrovela.com/)
and neighboring Farmers Market (http://www.farmersmarketla.com/) and a 3-mile drive to the both the Sunset Strip and central Hollywood. Here's a map:
Yes, this is a basic 'older 2-star' but it affords you and your sons some privacy and it's in a great area that isn't terribly touristy; at the same time, it's a short drive to all of your destinations.
The booking.com booking can be cancelled at no charge -- if you don't find something better. (Read the site's 'terms & conditions' regarding cancellation).
June is 'high season' so you'll need to book fairly soon in order to have the best choice. You haven't provided your exact travel dates, so we can't search for a deal for you. Prices will vary based upon the duration of your trip, whether it includes a weekend, etc.