Hedge your risk, not your cashflows
You don't need to square your book so that there is no net payment anywhere. You do need to hedge the interest rate risk.
Trying to match the fixed rate payments and ending up with a remainder on the floating side is not a good result; it replaces one dependence on future interest rates with a more complicated dependence.
The biggest risk is on the float leg payments; the fixed leg payments are known, so they are not at a great risk of fluctuating in value.
Ideally, match the notional structure of the swap
You can hedge the floating rate risk by matching the notionals on a series of swaps; suppose your swap notional has 2 steps from A->B and then B->C. Then we can use 3 swaps with notionals C (the lowest, running to maturity), B-C (covering the delta to the second step) and A-B (covering the delta to the first step).
There is a choice, then, on the fixed side of those swaps. Either we do them at the same fixed coupon as the original swap, and accept that we will pay or receive upfront for the nonzero present value, or we trade those at par for zero present value and accept that there will be a nonzero coupon balance at each cashflow date. In theory these are equivalent as you would expect to borrow/lend any upfront PV, but in practise you might have a view on which you prefer.
But you may need to compromise for liquidity
Hedging with liquid instruments (10y is much more liquid than 8y, for example) is generally more efficient as spreads are tighter. In tandem with that, the market usually hedges less liquid maturities using more liquid ones anyway (8y hedged with 5y and 10y) so it may prove operationally better to instead calculate your amortising swap's sensitivity to shifts in the key liquid swaps (e.g. 1, 2, 5, 10, 20, 30y), and hedge those sensitivities.
Such hedges (and even our ideal hedge) still need monitoring for their adequacy, but usually that is managed by looking at the whole book rather than individual swaps like this. It would, however, feature in the PV calculations for collateralisation.
Replacing a swap with one maturity with a set of swaps with different maturities can sometimes come undone with small differences in date schedules, particularly around ends of months, so care should be taken not to accidentally end up needing to cover the gap between one swap paying and another receiving.
Secondary interest rate risk
The secondary (much smaller) effect of interest rates is to change the discounting of the cashflows; a fixed leg still has a small value dependence on interest rates, so you may want to also cover that in some way. This kind of risk, however, is more easily managed by calculating an overall sensitivity and hedging that by adjusting the swap hedge slightly, rather than by trying to eliminate it entirely.
Beware principal repayments doing this cross currency
Unlike IRS, cross currency swaps usually trade with an exchange of principal. If your cross currency swap amortises, then there will be principal payments at some points in the schedule, which will need to match those in the hedging swaps. These payments are a much bigger deal than coupon cashflows, and forward exchanges are often not at par.