Why does changing the value a currency help/hurt exports if sellers can just inflate/deflate prices to match the change? 1) I make widgets and sell them for $10.

2) The government reduces the value of my currency by 10%.

3) The market value of my widgets is the same as before, so I raise my
prices to $11 and I can still sell them them because the (real) value hasn't changed.

Following this chain of reasoning means that devaluation of a currency wouldn't encourage exports, but that's not true.

An increase in the value of another currency relative to mine may allow you to buy more of MY dollars, but the market value of my goods is still the same.  The only people who would notice a difference when this happens would be those holding currency and not assets.  Are those people the target of currency devaluation/appreciation, or is there something wrong with my reasoning?
What is missing in your reasoning is the effect of devaluation on the seller's profit.

To keep things simple, assume that the inputs needed to produce the widgets are all bought domestically. Then costs will not change following devaluation.  But prices in terms of domestic currency can increase, without as you say increasing the cost to foreign buyers in terms of their currency.  So profits per widget - selling price less costs per widget, both in terms of domestic currency - can increase.

There is another possibility.  A seller might take advantage of devaluation to try to increase total profits by increasing the volume of exports. If so, they could increase the price in terms of domestic currency by less than 10%, or perhaps not at all, with the effect of reducing the price to foreign buyers.  Unless demand is perfectly inelastic, this will increase the volume of export sales and, if domestic sales are unchanged, the total volume of sales.  An increased production volume may in the long run permit economies of scale, reducing costs per widget.

Either of these strategies could yield an increase in total profits.  Which yields the larger increase will depend on both the elasticity of foreign demand and the scope for economies of scale. 

Through the profit motive, therefore, devaluation can certainly encourage firms to increase the value and/or volume of their exports.