Probably not, under the facts you describe. Slander and libel cases are reserved for areas that are really more black or white, and the manager's evaluation here is an entirely subjective thing. In Texas, most employment of this type is "at will," meaning that you can be fired for any reason or no reason, so long as it is not motivated by race, religion, sex, age, etc. Many people have been fired for lousy reasons, or false reasons, but trying to get monetary damages for that, or worse yet, trying to get your job back, is not something you should ever count on.
In Texas, most employee-employer relationships are "at will": Unless both sides have agreed in advance to guarantee employment for a particular period (or they've agreed to make continuation dependent on some other specific terms), then either employer or employee can end the relationship, in his/her/its sole discretion. It's hard to prove that there was such a guarantee unless it's in writing. And for everyone in a regular "at-will" employment relationship, it doesn't matter whether the employee had a good reason for quitting or the employer had a good reason for firing. No notice is required.
The consequence of this is that in general, there simply IS no case for "wrongful termination" in Texas. The exceptions are very, very rare -- e.g., if you can also prove that the sole reason for termination was your refusal to perform the employer's instruction to commit an illegal act, or if the specific intention behind the termination was to discriminate against you based on race, creed, sex, or national origin -- and these exceptions are very hard to prove.
Even if you sued the co-employee who you think libeled you instead of suing your employer, the defense would argue -- with considerable force -- that your lawsuit is really just a disguised attempt to get around Texas law's refusal to compensate for "wrongful termination." The co-employee would argue that what she said or wrote about you was an opinion -- which can't be sued on -- instead of a statement of fact. The co-employee would also probably argue that his/her own job requirements included assessing your performance, making her statements privileged. And of course he/she would argue truth -- that what he/she said or wrote was true, or close enough to true.
The defense would argue that losing your at-will job was no loss, since you had no reasonable expectation that the job would be around next week anyway, and that you have no damages (for the same reason). The defense would argue that you didn't try hard enough to find a good enough replacement job quickly enough, and that if you had, you would have sustained no damages. In any event, your damages (they will say) are speculative.
Not only your entire work history, but your entire life is likely to be put under a microscope, ostensibly because it relates to your damages and your credibility. It's not uncommon for libel or slander defendants to say: "I couldn't have harmed the plaintiff's reputation because she'd already ruined it herself."
Your ability to get "smoking gun" evidence from the employer's files would depend, ultimately, on whether they comply with their legal obligation to turn over stuff that incriminates them; dishonest employers sometimes shred records, and sometimes they get away with it.
If your case is brought against the co-employee, he or she may not have assets sufficient to pay the judgment you might win against her. Homeowners insurance might provide coverage, but those policies typically have an exclusion for work-related defamation claims.
The difficulties of winning the case, and of collecting if you win, may make it hard to find an attorney willing to gamble his time and expense-money under a contingent fee arrangement, so you might have to pay a lawyer cash in advance by the hour. Attorneys' fees aren't recoverable, though, so even if you won, you couldn't force the defendant to pay your legal fees.
There may be other important facts you haven't mentioned, and all I'm doing is pointing things out and making some educated guesses from what you have mentioned. Personally, I doubt your case makes economic sense to pursue.
If you do decide to pursue it, though, this is very important: The statute of limitations in Texas on libel and slander claims is ONE CALENDAR YEAR from the date of the publication of the false statement. If you don't sue within that year, you lose your rights to sue, period
One more quick point: I encourage you to consult with one or more attorneys directly, in a privileged setting of privacy -- in person, by phone, or by email, but not via a public website such as this one. If you're going to do that, do it right away.
You should check with a lawyer in Texas, but generally within the workplace there are limitations on defamation actions that include immunity for reports made in an employer-employee context, so long as those reports are for the purpose of hiring/firing/promotion and are not distributed outside the relevant company staff.
Sign up to receive a 3-part series of useful information and advice about personal injury law.