It is good that you have decided not to abscond and you must not.
Are you employed in India or Abroad?
It is guessed that:
1) 30 Days Unpaid is against which paid leave can be adjusted.
2) 60 Days Fully paid is period which is to be served by employee and is excluded of holidays.
However you may confirm what is clear implication of both.
In the appointment letter issued to you and accepted by you is there option of notice pay in lieu of termination of employment by employer and employee.
You can tender notice addressed to your appointing authority, with notice period up to a day before date of appointment in PSU (mention this date as effective date of resignation, and last day in office) and affirm to tender notice pay for the shortfall in notice period, either by a crossed a/c payee cheque favoring company (against proper receipt) or ask to adjust in FNF statement. If notice pay is not adjusted in FNF statement you may find it difficult to avail I tax relief. You may request the good offices to ensure smooth exit formalities within effective date of resignation and to confirm to whom you should handover the charge. You may ask for acknowledgment of notice of resignation immediately, acceptance of resignation, work experience/service certificate, relieving letter, form 16, settlement of a/c, disbursement of settlement dues, FNF statement, Pf number/accumulation reports/attested copy of PF withdrawal-transfer forms, NOC/NDC to be supplied to you within effective date of resignation. Ideally settlement and payment of wges should be given on last day in office or with 2 days.
The current employer mat become rigid an adamant, and may not issue relieving letter, therefore it shall be appropriate to communicate to concerned personnel in PSU that you shall be able to produce copy of notice of resignation, and proof of dispatch and delivery, last salary slip and your current employer may not provide acceptance of resignation, relieving letter and you shall be able to produce acceptance of resignation, relieving letter, if and only if and as and when your employer supplies it to you. Therefore you should be allowed to join and work on the strength of copy of notice of resignation, and proof of dispatch and delivery, last salary slip only.
These are the times when employee need to apply rapport, goodwill and exceptional levels of persuasion, persistence, reasoning, negotiation skills and resolve the matter while in employment and this shall be quickest and easiest solution. Your line management and HR can be meaningful help to you.