Am I overthinking my raised bed spring rejuvenation? Best practices?

So, I have 4 raised beds - 2 @18 inch and 2 @ 32 inch height.  Last year, I mulched with straw, which worked well, but the decomposition has been slow, to say the least.  A couple questions loom for me at the moment:
1.  I'd rather stay as close to a no-till planting approach as practical, so I'm wondering if it's a viable option to layer some of my Mel's mix over the top of the existing straw, or should I do a shallow turnover of the straw and top 1-2 inches of the bed before adding new growing mix on top?
2.  I don't have the option of taking the partially composted straw and throwing it in my barrel composter, because it's already full.
Thanks in advance!
Russ

My general approach is to weed the bed if necessary (or chop/drop if what's growing there isn't problematic). Then I temporarily rake off the mulch layer, topdress with a mixture of biochar and compost, sprinkle a bit of burned+crushed shells and bones, and then do a quick pass with the broadfork. After that, I level the surface and spread the mulch again, then plant into it. More wood chips get added during the growing season as plants get bigger.

Thanks, Phil!  This approach aligns with how I like to do things with minimal invasiveness.  We had quite a mild winter here around Seattle and you'd think that there would have been more breakdown, but that didn't happen.  I think we're already at the point where I can start planting my frost-tolerant brassicas and early lettuces, which is the earliest I've ever launched my season.

I like adding 6 inches to 12 inches of wood chips.  No sunlight can cause no weeds.

Over time the wood chips break down to soil and thus can be refreshed.

Adding mushrooms is a plus.

To plant. push back wood chips, dig a hole, drop in transplant.  

Win Win situation.

Hi Russell,
I would not mix the straw in at all. While it seems like this would be a good way to break down the straw faster... it can also cause problems with nitrogen availability. Plus its more work for not much benefit. I think it is better to leave the big stuff on top. Move it aside to add anything you need to add for compost and supplements, then you can put it back. It will break down eventually, but in the mean time it can act as mulch and provide those benefits.