You may ask the school system for more comprehensive testing if you have a disability-related reason. Otherwise, the district is not obligated to assess your children. In your DS's case, it sounds like you suspect a language disability. In that case, you could write a letter to the director of special education requesting a comprehensive evaluation, including at least cognitive, speech and language, and achievement areas, to address a suspected language disability. Depending on the state, they may be able to refuse, but will have to state a reason to do so, and what action they propose instead. Either way, they must respond in a timely manner (typically 5-10 school days). In your DD's case, you do not report any areas of concern that would be relevant to a free, appropriate public education, so there is no basis for making a request for special education evaluation (which is pretty much the only way to get comprehensive testing out of the public schools, if they don't offer it as part of GT program selection).
If you choose to cold-call the nearest university, you may wish to consider the kind of psych department they have. An experimental psych program is not what you need, in this case. What you are looking for is a doctoral-level clinical or counseling psych program, or a master's, specialist, or doctoral-level school or educational psych program. If there is a neuropsych doctoral internship attached to the uni, that would also be excellent.
If there is a children's hospital nearby, there may also be a learning disorders unit there, which might work for a neuropsych for your son.
Did you try the state gifted child association? They might have additional referrals.
You may also be able to find school-employed psychs in neighboring districts, or other private psychs, with small private practices. (There are ethical questions if psychs employed by your district evaluate your children for private or insurance compensation, since, hypothetically, they could be called on to assess your children using taxpayer funding, in their formal employment role.) Sometimes you can find these individuals by calling up your school district and asking if they have recommendations for private evaluators. Every school system has a short list of private evaluators that they maintain for obtaining Independent Educational Evaluations when requested by parents through the IEP process.
