Since I've joined the crypto revolution, I've been accumulating Bitcoin as a sort of retirement fund. My cold storage wallet has been intact for nearly 6 years now. Thanks to the "fork craze" of 2017, my money has quadrupled. Talk about crypto being better than banks.
I don't know how you made that math, specially since you didn't even claimed your forked coins now. BCH and BSV are in a strong downtrend and they will soon be worth nothing.
BTC/BCH ATH ratio was 0.5, now it is 0.011
You certainly aren't going to quadruplicate your BTCs, you will, at the most, get 0.015 of each btc claiming all forked shitcoins.
Now, I want to move my "Pre-Forked" Bitcoin funds to a new address. But I don't know how to do it without risking losing my funds from other blockchain networks.
All I want to do is send my "Pre-Forked" Bitcoin to the new native SegWit (Bech32) cold storage wallet, while "splitting" forked coins into new addresses.Can you help me?

I would transfer my BTC to another address first. Just buy a Hardware wallet (since you have many btc from many years old, 80 usd won't be a problem). I recommend Ledger Nano.
Then send your bitcoins to a native segwit addres
Then, you should follow this guide, which is the best imo:
https://coinomi.freshdesk.com/support/solutions/articles/29000026274-bch-abc-bsv-fork-information-splitting