What you have there is simply water seeping out of the gel and bringing some dissolved stuff with it. This is known technically as syneresis. What will help is to add something stabilize the gel. Xanthan gum is probably the easiest thing to use. You can find it at health food stores or Whole Foods because gluten-free bakers use it a lot. Start with 1/8 teaspoon pureed into your filling. Sprinkle it over the filling liquid before mixing, and put it through a sieve before baking to catch any clumps.
Almost any normal sorbet recipe will contain a decent amount of sugar, and strawberries are no exception. I'd guess probably 1/2-2/3 cup per pound of strawberries. Use a substitute if you have an aversion - honey, agave, raw cane sugar, whatever you prefer. (Of course, anything liquid is going to contain some water, and cause a bit of ice, but it's still doable.) A sorbet without any extra sugar, even when frozen normally in an ice cream maker, is going to end up with a very icy texture, which will probably keep you from experiencing the flavor as much. The sugar helps soften it.
The other common way to soften a sorbet is alcohol. Rosé wine (thank you, David Lebovitz) works quite well in a strawberry sorbet. Vodka can be your go-to liquor for any impromptu sorbet with other fruits, since it'll add alcohol without any flavors that clash. Beyond that, either look for recipes, or look for mixed drinks containing the fruit you're using.
You might also be having problems simply because your strawberries aren't that great. Did you freeze them, or were they storebought? If you bought the strawberries yourself, you'd know they were decent before freezing, while not all storebought frozen fruit is as flavorful.
Finally, it's possible that your frozen strawberries collected some extra water in the form of frost, which then makes your sorbet icier. If there are obvious big chunks you could scrape them off.
Best Answer
No it won't be a problem.
For safe canning, time, temperature and sufficient acidity are the relevant variables. Neither is affected by your substitution.
Taste-wise switching from lemon to lime will change the results, but obviously that is exactly your goal. (Nice idea, that "Strawberry Margarita preserve" btw.)