Some subtle clues in text that Severus Snape felt ashamed of being a DE in past.
#1. Goblet of Fire - The Egg and the Eye
‘Course Dumbledore trusts you,’ growled Moody. ‘He’s a trusting man,
isn’t he? Believes in second chances. But me – I say there are spots that don’t
come off, Snape. Spots that never come off, d’you know what I mean?’
Snape suddenly did something very strange. He seized his left forearm
convulsively with his right hand, as though something on it had hurt him.
Moody laughed. ‘Get back to bed, Snape.’
‘You don’t have the authority to send me anywhere!’ Snape hissed, letting
go of his arm as though angry with himself.
(See, he feels ashamed of having that Dark Mark, he feels “angry with himself” for being a DE in the past.)
#2. Deathly Hallows - The Prince’s Tale. (although HBP’s timeline)
“Information,” repeated Snape. “You trust him … you do not trust me.”
“It is not a question of trust. I have, as we both know, limited time. It is
essential that I give the boy enough information for him to do what he needs
to do.”
“And why may I not have the same information?”, asked Snape.
‘I prefer not to put all of my secrets in one basket, particularly not a basket
that spends so much time dangling on the arm of Lord Voldemort.’, said Dumbledore
‘Which I do on your orders!’
( He felt he deserved to know about the information because he was a true Order member at heart. He doesn't want to associate with Voldemort at all but he has to do it only because Dumbledore wants him to spy on Voldemort. He does not want to “dangle on the arms of Voldemort” even as a spy. He wants to completely dissociate himself from DEs.)
#3. Deathly Hallows - Sacking of Severus Snape
‘I was under the impression,’ said Snape, ‘that Alecto had apprehended an
intruder.’
‘Really?’ said Professor McGonagall. ‘And what gave you that
impression?’
Snape made a slight flexing movement of his left arm, where the Dark
Mark was branded into his skin.
‘Oh, but naturally,’ said Professor McGonagall. ‘You Death Eaters have
your own private means of communication, I forgot.’
( He is so ashamed of having that Dark Mark that he just merely hints at it in a very subtle and physical way rather than acknowledging it with words. Up till that point, he was answering in words.)
These are three of the instances that indicate how much Severus Snape, after change of sides, loathed to be even associated with Voldemort and his Death Eaters, to a point where he felt ashamed to acknowledge his Dark Mark! He merely hinted at his Dark Mark, in a very subtle and physical way, rather than flaunt it using words. How much Severus didn’t want to “dangle on the arm of Lord Voldemort” even as a spy but he didn’t really have much of a choice, did he? —
“Well, you don’t just hand in your resignation to Voldemort. It’s either a lifetime of service or death”.
To Severus, a third option - being a double agent spy pretending to be Voldemort’s supporter whilst actually secretly working against him - was presented by Dumbledore, which he agreed to do not once but twice (1st war and 2nd war).
But Severus was a human, afterall. Even the pretense of supporting the evil could be very taxing and it’s evident from the above three instances. You feel sorry for this man but you also know that he was, afterall, paying the price of his own terrible mistake that he commited around couple of decades back as a teenager. You feel sorry for him, but you also feel appreciative of the fact that this man was willing to do and sacrifice anything and everything to atone for his mistake, that he never made excuses or ever backed off from the difficult road of atonement, and in the end, paid even the ultimate price — his life.