I don’t do much heavy lifting or digging anymore either.
Since this is your own flower bed and you know which plants you want to save, I think the easiest method would be to lift those plants first, although the later-emerging hosta might be a problem in winter/early spring unless there are dead leaves to show where they are.
Many weed killers are best applied when the weeds are in active growth, so they will absorb the chemical. Anything that will kill them in the soil are likely to obstruct/poison future plantings in the flower bed.
I think immediate action would be to heavily cardboard the weedy part of the bed — this will smother any weed seeds trying to grow, and weeds trying to re-emerge when the ground thaws. Cut any standing weeds to the ground with loppers or saw if necessary and leave the dead debris under the cardboard to decay and become part of the soil — the debris should be spread evenly. BUT if you suspect diseased plants, then any dead plant material should be removed, bagged, and put in the trash if possible (or use your local yard waste collection method).
For the time being, you can leave the hosta and sedum sections uncovered, but spread any soil mineral amendments (lime, phosphate, etc.) and composted manure, etc. UNDERNEATH/BEFORE putting down the cardboard. Personally, I would just lay bags of mulch on top of the cardboard to weigh down until spring — mulch to be used later after all the work is finished.
Once the ground thaws and temperature is warmed up enough to work on the retainer wall — which would require clearing and digging out the surrounding edge of the bed — the weeds along the border of the cardboard can be dug out in the process, and the desirable plants can be lifted and temporarily preserved (I like putting the rootball in tripled plastic grocery bags for temporary storage). Then lay the additional wall blocks, and maybe lay another layer of cardboard. Then the thoroughly cardboarded bed should be watered well to soak the cardboard.
The addition of 6 inches of good weed-free raised bed soil on top should be sufficient to support the sedum and hosta if holes are cut into the cardboard for their roots. You could plant shallower rooted bedding annuals in the 6 inches to enjoy the rest of the flower bed for this season. You could also set down container plants nestled in the soil and mulch.
By next year, existing weeds will be smothered and the cardboard should be on their way to breaking down into the soil. Usually, earthworms will have gathered and will have had a big party in there, helping to loosen up and invigorate the soil.
— if there are REALLY aggressive perennial weeds in there, they might survive being smothered for the year, or they might find their way to the holes for the sedum and hosta, but you could keep cutting them to the ground to weaken them — no leaves, no photosynthesis, and they will use up their root-stored energy.
...I couldn’t tell you if there are any organizations... I suppose though, for this much work, you could hire neighborhood teens or a local handyman. They would need to be supervised about which plants to keep and how to lift them, but otherwise should be able to do the work?
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