I have a few raised beds and the same problem. I would re-use what you have in the beds and top-dress it, because that allows the soil to develop. Your earthworms and other buggies will make sure the stuff mixes. In my really tall raised bed, as filler I shoved strawbales in the bottom. It's NOT good for nutrients (initially, must decompose first) and so I had to compensate with fertilizer (blood meal), but when your raised bed is 3 ft high (to prevent/discourage bunny entrance), it worked! Also, I add brown paper grocery bags, recycled/bleach-free paper towels, empty toilet paper rolls, and the occasional paper napkin to my compost and try to compensate with nitrogen rich other stuff (which is usually our leftovers) to make sure we strike the right balance. Sometimes you can ask tree trimmers for wood chips. In my neighborhood, you can gather your neighbor's lawn bags or leaves. In some places the local waste treatment plant has compost....of course, you're not asking about that. They say no putting of dog poop or humanure in composts because of the threat of pathogens. I have a few wild areas that I hack at every few months and gather the debris for compost. It comes back. There are also a few perennials you can grow for the occasional food and biomass, that are good nutrient accumulators. I use horse radish because we like it occasionally, it's perennial, pulls nutrients from deep, and I have no qualms pulling off a few leaves here and there to get some extra biomass.
I actually have the exact same problem. I have 96 sq ft. of 15" high raised beds on my small suburban plot and the soil volume lost year-to-year is about the same amount of soil I make from my own yard waste. This'll be a problem when I add another 24 - 40 sq ft of bed next year. Sand is great for drainage, though I've never had drainage problems with my compost-enriched soil. Perlite does the same at the cost of money and being non-renewable, so I only use it to fluff up my sterile seed-starting mix (considering substituting coir). Here are a few ideas off the top of my head:
Bury brush/sticks/whatever organic yard waste you can't process easily for compost like a micro-hugel
Grow cover crops throughout the year and chop/drop as needed. Throw these on the pile or compost in-place. I'm thinking of trying some hairy vetch and winter pea this fall
Mix desired volume of clean soil from elsewhere on your property into the compost. You'll increase biological diversity in the compost and kill latent weed seeds while you're at it
Find clean, external sources of compostable material-- Amit (howdy, fellow Ohioan!) mentioned lawn waste but let your imagination run wild: coffee grounds and food scraps from a local cafe or diner, spoiled straw or hay (especially alfalfa!) from local farmers, damaged produce from the farmer's market, stale bread from bakeries, wood scraps from the local firewood guy, sawdust from the local sawmill
seasonal compostables (there are a LOT of rotten pumpkins around after Oct 31st if your pile isn't frozen)
To increase the quantity of compostables for your heaps, check with neighbors that don't compost, they may be happy to give you their leaves, grass cuttings, etc.
You might also check with local stores, many of them will have boxes to get rid of as well as other paper goods.
Don't forget to ask the tree trimmers if they need a free spot to dump their chipped up limbs, this can be a Woot! moment if they do.
As has been mentioned using your compost as a top dressing saves on the quantity of compost you need.
I top dress with anything and everything right now trying to bring my dead dirt back to life. Lots and lots of paper and if I have to, I throw some alfalfa meal on top to balance things. I just need mass and nutrients which I know the worms and other microherds can help with. I dig holes a couple feet deep all around the yard and just start piling stuff in.
My goal? To create a wildlife habitat on our surburban (almost rural) property using mostly California native plants and inspire others to do the same.
Also to learn how to grow some edibles - more than the herbs I have grown before.
Roses are red, violets are blue. Some poems rhyme and some don't. And some poems are a tiny ad.