You cannot be illegally retained by your current employer in the following cases:
1) If you have served the notice period in writing as per your current company policy.
2) If you have handed over your charge to the concerned authority or at least asked them to take the charge in writing.
3) If you have intended to complete the exit formalities in writing to your reporting head.
I emphasize using "in writing" to ensure that if they do not release you on time, do not provide a relieving letter, or have pending dues, you can retain the written documents as evidence to legally claim them by approaching the local governing body for disputes.
Presenting your offer letter will not be an issue unless your manager engages in unethical behavior, such as providing false information to the new company about your performance, behavior, or other aspects.
Another way to address this issue is to speak with the HR of the new company, express your willingness to join, and explain how your past company is causing issues with your relieving process. If you are transparent with them and have completed all your formalities with your current company, the new HR will likely assist you in resolving the matter.
Hope this information is helpful.
Best regards