First of all, there is no 'legacy blockchain' and no 'segwit blockchain'. It is bitcoin, it is one blockchain.
You can 'store your coins' in either 1) a legacy address or 2) an address in segwit format.
The fees of BTC are going higher if more people want to send coins around and are ready to pay more for that.
And if people decide to not send too much across the BTC network (and are not ready to pay 30$ for a TX), the median price of a transaction will go down.
When sending from an segwit address you can save about 35% of the size of the TX (and therefore from fee, since fee is usually calculated in sat/B).
It doesn't matter what address (P2PKH/P2SH) is recieving your transaction.