You are right to be suspicious. If you in fact do not have HDHP coverage (it sounds like you don't, but your benefits coordinator or the insurance representative would know for sure), then the annual limit for your wife's HSA is $3,400 (assuming that she is under 55). Note that there are more requirements to be an HDHP policy than just have high deductibles.
The question of coverage is totally a function of what the actual HDHP insurance is (and the only two choices are "Self-only" and "Family"). Your filing status has nothing to do with this. It is entirely possible for two married people to have their own Self-only HDHP plans and their own HSAs, in which case, each of them gets a $3,400 limit (but no more than $3,400 in any one HSA).
Your university plan being "Self-only" is not "Self-only" for purposes of an HDHP or HSA; it just covers only you, that all.
Your wife contributed $3,400 to her HSA - that's great. But unless you discover that your health policy really is an HDHP-qualified plan, that's all you can do.
To boost your contribution limit, you would have to change your wife's plan to Family coverage. This is certainly doable, although I don't know what the financial impact would be.
NOTE: this is odd, but your wife can actually boost her limit to $6,750 if she is on a Family plan, even if you are not eligible to contribute because you have conflicting coverage. The IRS simply requires that your wife have one other person on the policy to have Family coverage, even if that person is otherwise not-HSA eligible.
Perhaps this is what they were thinking at the seminar.
And if you drop your university coverage as well as join her on the Family plan, then not only is your limit now $6,750, but YOU can also contribute to her HSA. This might make the whole process more feasible.
Last note: if you make the switch mid-year, note that if your spouse has Family HDHP coverage on December 1, 2018 (or whatever year), then she is allowed to contribute to full Family amount, even if she didn't have the Family policy all year (the annual HSA contribution limit is the aggregate of each month's limit). This is called the "last-month" rule. The catch is that she would have to stay under HDHP coverage (this one or another if she changed jobs) for the entire next year or face penalties...nothing is free...
