The simplest thing would be to ask your former employer to sign the form certifying the dates that you worked for him/her. This happens frequently, signing off on a former employee's work. Unless you left on bad terms, I'd just go back to them and say “Hey, I've completed my exams and am working through the licensure process – can you please complete this form for me? Thanks!” Shouldn't be an issue. If you did leave on bad terms, then the process gets complicated.
Some states require an employer to complete the forms, and will pursue disciplinary action against the employer and his/her license if s/he won't. Other states leave it solely up to the discretion of the employer. This usually varies based on whether the employer's form is more of a “verification of employment”, or a statement regarding the abilities acquired by the employee. If it's just “Jane Doe worked for me from March 2013 to June 2016”, then it's more likely to be required; if it's “Jane Doe worked for me from March 2013 to June 2016 and during this time acquired the skills necessary for licensure as a CPA and performed well in executing her duties in these areas”, then it's likely to be at the discretion of the employer. Given that Texas's statement is more along the lines of “This individual did well and was moral and upright while I supervised her”, I can't see Texas forcing someone to sign it, since it might require someone to sign something they didn't agree with.
So, a state board does have ways to verify whether or not you worked somewhere, but without being able to get that signature, it's not likely to count for your experience verification, unless you're in a state which mandates the employers complete it, and I'd doubt Texas is one of those. Hopefully you parted on good terms with your prior employer and can just simply request the form be completed. You probably can start the process now if it's concerning you – just tell them you're nearing completion of your exams so wanted to send them the forms now so that they could complete them at their convenience, and that way, if they're opposed to it, you'd know sooner rather than later.
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