What is the term for having dyslexic sorts of issues with spoken words?

Dyslexia is defined as

... a specific learning disability that is neurobiological in origin. It is characterized by difficulties with accurate and/or fluent word recognition and by poor spelling and decoding abilities. These difficulties typically result from a deficit in the phonological component of language that is often unexpected in relation to other cognitive abilities and the provision of effective classroom instruction.

It's frequently popularized as just scrambling the letters of words, but the dyslexiaida.org site is very clear that it is not limited to that.

But my question is about speaking difficulties. Dyslexiaida.org states

People with dyslexia can also have problems with spoken language,

however, reading further on the site, it is at least strongly implied that this mostly pertains to difficulties reading aloud and difficulties with pronunciation that were acquired due to errors in reading.

Example sentence:

Many years ago, I accidentally referred to myself as the wife of my then-wife, rather than her husband, due to _________.

Generally, when I experience this, I will think the right word, but say the wrong word. The word spoken will often be related in some way to the word desired.

Note that this is not always what's going on in my head when I get left and right mixed up. As a left-handed person, having been taught the mnemonic "Your right hand is the hand you write with." even 40 years later I'll sometimes genuinely get them mixed up. I don't care if the word covers this class of mistake, too, or not.

Possibly related would be the tendency to sometimes accidentally rearrange syllables. While I'd be interested to know this, I'm not nearly as interested, as it's a very rare occurrence in the people with whom I interact. (Example: one time I said "lexdysia" when I meant to say "dyslexia". The first time I attempted to share about that issue, I accidentally said "dyslexia" when I intended to deliberately say it wrong again.)

While I am American, I have a preference for being able to communicate with people from all over the world as much as is feasible for a hopelessly monolinguistic person such as myself, so I'm not using country-specific tags.

Notes: dysarthria is difficulty speaking due to brain damage or changes later in life. It's also a muscular thing, rather than using the wrong word. The difficulties that I'm asking about in this post have been things that the people who struggle with them have had to deal with their whole lives.

Dystonia is difficulty speaking due to muscle control issues.

Dysprosody is a false accent.

Aphasia is, among other things, the inability to find the right word to use. In the case I'm talking about, the right word is known, and even thought, just not said.

I would think this would fall in the range of semantic disorders, but it's pretty distinct from the most common semantic disorders listed.

Speech difficulty is much too general, that's all of these things.

This sounds like a very mild case of verbal paraphasia. From Wikipedia:

Verbal paraphasias are confusions of words or the replacement of one word by another real word . . .

You mention that

The word spoken will often be related in some way to the word desired.

That matches one aspect of verbal paraphasia:

These errors can be semantic, in which the meaning of the word is related to that of the intended word

There are a variety of ways in which the terms can be related; your wife/husband example might be a coordinate semantic paraphasia, in which "the target word [is replaced] with one that is from the same category." Other examples from the article (each with its own subcategory) include shoe or leg in place of foot; fruit in place of pear or vice versa; and knife for nail (because knives and nails are visually similar).

Paraphasia is generally associated with aphasia and thus, as you mention, with brain damage of some sort. However, the Wikipedia article notes that verbal paraphasia associations are a bit different, and

It is hypothesized that verbal paraphasias are not the result of a random process but from a precise deficit in a single area.

I'm not a neuroscientist, but it seems at least plausible that some organic issue in that "single area" could result in the occasional speech misfire that you've described.

This all depends on what exactly you have in mind. Saying "wife" instead of husband or mixing up words like that I would describe as a "slip". There's a Wikipedia article on speech error, which also calls it a "slip of the tongue" or simply "misspeaking".

In the speech error article of Wikipedia it lists a number of examples you may have in mind:

Spoonerism

Target: I saw you light a fire.
Error: I saw you fight a liar.

Substitution

Target: Where is my tennis racquet?
Error: Where is my tennis bat?

Word-exchange error

Target: I must let the cat out of the house.
Error: I must let the house out of the cat.

The article also gives the following as a substitution error:

Target: George’s wife
Error: George’s life

There are a few more in that speech error list, I'm not exactly sure what you're looking for, or what category your wife/husband example falls in, if any.

There's also something called speech disfluency, which I suppose is even more broad. The article includes slips of the tongue within this category. The reason I bring this up is that you mentioned dyslexia and when you spoke about the verbal version of it this is what came to mind, but I may completely have the wrong concept of it.

Possibly related would be the tendency to sometimes accidentally rearrange syllables.

Spoonerisms are of this type:

"The Lord is a shoving leopard" instead of "The Lord is a loving shepherd."
Spoonerism

Sometimes syllables and morphemes are mixed up, sometimes words. That speech error article and its links cover quite a bit.

So in your example sentence I would probably say a slip, or slip of the tongue if I wanted to be more specific (though I don't think I'd need to in the context).

slip
3.b.A slight error or oversight, as in speech or writing:
a slip of the tongue.
American Heritage Dictionary