If fees go up to just a few sat/byte, it will cost more to spend the dust than it's worth.
Some wallets allow locking unspent inputs, Bitcoin Core for instance gives this option:
Option > Wallet > Enable coin control features > OK
Send > Inputs > "right mouse" on an input > Lock unspent.
In Electrum it is very simple to do it. Just click in the Coin tab and choose which UTXO you want to spend (each of those dusts are going to be one UTXO individually).
It´s probably not worth claiming them, if there are too many transactions.
Maybe the best option would be to stop using that address and wallet.
Someone is tracking you. I think it would be a case to use a
coin Mixer. Send all your funds from that wallet/address to a mixer a then move it to a fresh new wallet.