Your edit gets at the heart of the matter. When you enter a forward contract it generally should be very close to net zero current value. So credit risk for either side is negligible and should the counterparty immediately default one can likely reenter the desired position with a second counterparty near the desired strike so future uncertainty not a large concern either at least if the market is liquid.
However, as time progresses your contract may become significantly in the money. In this case the larger amount owed to you increases the counterparty risk at the same time it increases the chance the counterparty will default as they owe more money in general. Also, if the position has moved in your favor it is now very hard to take on a contract at a similar strike if the counterparty does default. This can be devastating if you were counting on these forwards as a hedge.
So while the odds of a counterparty defaulting during the period of the contract decrease over time the pain from a default can increase significantly. Many forwards contracts will have some (usually banded) mark-to-market accounting to mitigate some of this risk.