In this situation the object of to refers almost exclusively to who the subject is married to.
A Corpus of Contemporary American English search turns up 5778 collocations of "married to," and all of the ones I've looked at identify a married couple. "I got married to Priscilla" would mean that you and Priscilla became spouses. This is the normative advice given by the Cambridge Dictionary, which says:
We use to, not with, after get married + direct object and be married + direct object:
Why do they recommend that? "with" tends to be used in reference to other people involved in a marriage.
First, let's take the more general case of "Married with." (We'll get lots of results with this to see how the preposition works.) "Married with" has 629 results in COCA. Let me show you a sample page of results:
They all refer to marriage with kids or children of varying numbers! This obviously doesn't mean that the subject of the sentence has become the spouse of one or more kids. Instead, with seems to denote someone who is related to that relationship more broadly -
He's happily married with kids
meaning he's (a) married and (b) with kids, or that the kids are a product or an accompaniment of the marriage.
Outside of these results, with can signify several relationships. Kids, of course, may be implied. So might a spouse:
1997 SPOK I met [Princess Diana] really at the beginning when she was married with Prince Charles.
2012 MAG I'm married with dogs, no children
Most of these functions carry over to "get married with" in various tenses and forms of to get. I found 21 results for "GET married with" (the capitalization generates all the verb forms of get -results came up for get, got, getting, and gets.) Note this compares to 274 results for "GET married to" - this is already a less common usage.
Similar categories of prepositional object recur with this more specific usage. There are people who are not the marital partner:
it's perfectly fine to get married with just your family and your very closest friends.
To get married with your child in your arms ...
So might one's current condition or possession:
Two days later, at age 22, Griffin got married with a black eye.
she's getting married with lipstick and alcohol and a skirt that's knee length
Everyone gets married with four sets [of linens] these days
So might a circumstance:
A girl can get married with parental permission at 16
It'll be cool to get married with all the fish swimming around us...
Of course, all this is in addition to a marital partner:
You know, I'd like to get married with my girlfriend.
With all of these objects for with, the personal ones contain an ambiguity even in "get married with" - it may refer to people at the wedding or it may refer to the marital partner at the wedding. It's also theoretically possible for a person to be part of a circumstance. With this potential ambiguity, that is likely why Cambridge Dictionary recommends avoiding it for the more clear (and much more often used) "get married to."
So "get married to" is a preferred collocation when referring to a spouse. The object of "get married with" would be determined more by context, and less commonly refers to a spouse.