Texas has a new due date for this year of June 15, 2021 but can I still get an extension?

You are too late to e-file an extension request   You could only e-file an extension request up through May 17.   Now the only way to request an extension (if you live in a state affected by the winter storm) is by mailing it by midnight tonight.   An extension will not give you extra time to pay tax owed.  It gives you extra time to prepare and file your Form 1040.

https://www.irs.gov/pub/irs-pdf/f4868.pdf

you have to mail it in and have it postmarked before midnight on 6/15/2021. the IRS closed e-filing of extensions on 5/17/2021

Victims in FEMA Disaster Areas: Mail Your Request for an Extension of Time to File
To request an extension to file your federal taxes after May 17, 2021, print and mail Form 4868, Application for Automatic Extension of Time To File U.S. Individual Income Tax Return. We can’t process extension requests filed electronically after May 17, 2021.

Find out where to mail your form.

Please be aware that:

An extension of time to file your return does not grant you any extension of time to pay your taxes.
You should estimate and pay any owed taxes by your regular deadline to help avoid possible penalties.
You must file your extension request no later than the regular due date of your return.
Filing this form timely gives you until October 15 to file a return.
To get the extension, you must estimate your tax liability on this form and should also pay any amount due.
Get an extension when you make a payment
You can also get an extension by paying all or part of your estimated income tax due and indicate that the payment is for an extension using Direct Pay or a credit or debit card. This way you won’t have to file a separate extension form and you will receive a confirmation number for your records.

Yes you can.   They only moved back the date to file without an extension request.