Yes, some numerals have a shorter form, used in day-to-day speech. It is important to note that these are mostly changes in speech and are rarely reflected in writing. And since Romanian has a pretty good letters-sounds correspondence, I will use letters to refer to both.
For numbers 11-19, "-sprezece" becomes "-șpe":
doișpe - although the form "douășpe" exists and it is used, very often people will not bother with the agreement and will simply use "doișpe" (e.g. "doișpe oi", "ora doișpe"). It's a syllable shorter.
cinșpe - the second "ci" in the long form "cincisprezece" is not pronounced either. I've rarely seen the written form "cincișpe".
For numbers larger than twenty, "-zeci" becomes either "-ș" or "-j" (depending on the following sound), but never when they are on their own.
I.e. you can say:
Am treij de oi. - I have thirty sheep.
Câte oi ai? Treij două. - How many sheep do you have? Thirty-two.
But you can't say:
Câte oi ai? Treij.
With the exception of "cincizeci" and "optzeci" which would be impossible to pronounce, thus they keep an additional "ze":
Am cincizej două de oi. - I have fifty-two sheep.
Am optzej două de oi. - I have eighty-two sheep.
"-ș" is used before an unvoiced consonant, and "-j" before a voiced one:
Treij două de oi. - Thirty-two sheep.
Treiș trei de oi. - Thirty-three sheep.
Keep in mind that these are rarely written down, but some people might use either "-ș" or "-j" exclusively, regardless of the next sound, since the voiced/unvoiced distinction comes natural anyway.
As far as I know (I could be wrong), there are no "official" guidelines on these (e.g. "treij două" vs. "treijdouă" vs. "treij-două").
* - this is the form as it appears in DEX. However, many people pronounce it "șaptișpe" (and, similarly, "șaptisprezece").
** - weirdly enough, the DEX does not list a short form for "optsprezece", but I guarantee it exists. First of all, there are two alternative pronunciations of the long form, both adding a helping vowel: "optisprezece" and "optâsprezece", thus the short forms "optișpe" and "optâșpe". I'd say that "optșpe" is not pronounceable.