Assuming the trees are planted in a good position and are where you want them. If you know how to graft and have a source for bud wood, that would be a nice option. chop the tree down right at the graft and let the root stock sprout some new growth and graft to that.
Truthfully though, just starting with a new tree will probably get you about the same amount of fruit in about the same about of time.

Andrea perhaps there is a 3rd option that may save your trees and make them produce fruit.

Think about the difference in temperature between shade and full sun during the winter months. Think about how much warmer the south facing wall of your house is verses the north facing shaded side.

If you could build a sun barrier for them they would get more chill hours; perhaps more than enough for them to fruit.
A couple of fence posts with a 2"X4" running on the top and bottom with a removable barrier could give them more chill hours. The barrier could be anything that blocks full sun. Something like 100% shade cloth or 1"X6" boards inside a 2"X4" removable frame.
The 1"X6"'s could held in the removable frame with 1"X2"'s on the top and bottom of the frame. The removable frame could be secured to the 2"X4"'s that are connected to the posts with screws so they could be removed each spring.

Another option would be to dig them up and move them to a spot that is shaded during the winter months. Not an easy task with six year old trees but could be done.

The answer is straight forward: top work your trees. It is not true that planting a new tree will give you fruit faster. On the contrary, if you top work the tree with low chill varieties, you will get fruit the following year. I just top worked 5 peaches last Spring. In one season, each graft grew 5 feet and even branched out. This year, exactly one year later, they are covered with blooms and should fruit nicely.

I am surprised that flavor queen is not fruiting for you, it's not a very high chill variety, mine flowers even after a short period of drought, i.e. I can get a Fall flowering if I withhold water in late Summer, and it is one of the first varieties to emerge out of dormancy at the first warm Spell in late January, blooming only a couple of weeks after Anna and Dorsett golden, so I suspect this variety needs next to nothing chill-wise. It's probably not 500 hours, because it fruits well in areas in the Monterey Bay that get under 400 hours of chill.

I would hold on to the Flavor Queen, it tends to not be very vigorous, something else might be going on. But you can safely top work the apricot to a lower chill variety like katy.

BTW, I have a similar problem with katy, barely any blooms on the tree two years in a row now, and it's supposed to be a low chill variety. (we have 600-1000 hours of chill annually, the low end is during El Nino years like this one) ) I don't know what is going on, I suspect it's getting too much fertilizer.

I am curious what other varieties you have that do well for you. That would give me a better feel for how much chill you are really getting. If you are so close to the ocean that it rarely drops below 45F, then you might have a chill issue with Flavor Queen.

Thanks for all of the thoughtful suggestions. As the crow flies I live 9 miles from the coast in zone 10. Here is a list of my trees that are doing well. The highest chill hour tree I have is the Arctic Star Nectarine at 400hrs.

Arctic Star Nectarine 400hrs
Panamint Nectarine 250hrs
Santa Rosa Plum 300hrs
4 in 1 Peach (June Pride, Evas Pride, Mid Pride, August Pride) 200hrs
Gold Nugget Loquat
Black Mission Fig 100hrs
Fuyu Persimmon (Imoto Strain) 200hrs
Coffeecake Persimmon 200hrs
Pomegranate Wonderful 150hrs

If I can graft low chill varities on top of the Flavor Queen that would be great. I just wasn't sure if that would work. I'm getting 500 hrs for the Flavor Queen off of the Dave Wilson tag that came with the tree.

Someone suggested throwing ice around the drip line for a few nights in a row to achieve the required chill. Has anyone heard of that method?

Also, all of my trees are in a single line on a hill at the back of my property which faces North. There is an 8ft wooden fence about 6ft behind them and the trees are planted about 4ft on center apart as described on the Dave Wilson website. I hope this info helps. Thanks!

andrea, I have Beauty and Methley plum trees still dormant. Each needs only 250 chill hours. Use e-mail.