I'm personally still in the re-fi camp.
OP has ~$1200 expenditures under current situation (before any cuts) and ~$910 of debt service and takes home ~$2000 / month (before parental loan payoff assistance).
I calculate based on the payment amount and rate the existing loans have about a 10 yr weighted average term, so over the next 10 years w/ no extra payments, he/she will pay about $18K of interest, ~$1,500 / year, it starts around $3,400 / years and goes down like any amortizing loan.<---let me know if I'm off here, I'm just using a straight 3.8% rate rather than giving the benefit of paying the higher cost loans off first, so it won't be perfect.
If OP extends to 20 years at 5.4%, payment goes to ~$614, total interest over the course of the loan w/ no extra payments goes to $57K (~$2,800) / year, starting at $4,800 / year run rate and decreasing thereafter.
that $40K difference in incremental total interest paid is a big scary number, but not if it frees up ~$3,500 / year of cash flow over the next ten years ($35K), and offers greater overall flexibility. Plus that $40K of extra interest is a the maximum that would be paid and doesn't take into account any job changes which are highly likely. Extra principal payments (from parents, and raises) will decrease the payment further to the point where it becomes but a footnote in his/her budget.
OP has an amortization / cash flow problem and an income problem moreso than an expense / interest problem. Extending term helps the cash flow problem and is a Band-Aid until the income problem (new/better job offer in hand) is solved.
I don't disagree with everyone that if OP gets a new more sustainable job situation TOMMOROW that the lower rate quicker amort loans should be kept in place, I would simply value the extra breathing room. AND not to mention the whole dynamic where the parental assistance covers a greater portion of the debt burden when the payment is lower. The flip side / happier version of that is the lower payment makes it potentially easier for OP to NOT have to rely on that $400/month. If some cuts were made, the whole debt service could be covered from the meager take-home pay at a $600/motnh payment more so than a $900 / month payment.
In short, I understand why people think it's bad advice to extend, but I think the benefits outweigh the costs.
Also OP. The state of new jersey will pay you back your contributions to the pension when you leave (with interest if you stay for 3 years). that's $3,100 / year (trapped in a retirement account) that you are saving that you aren't keeping track of/aren't seeing, so that's nice from a mental perspective.
EDIT:
Just to further elaborate here, if you made minimum $900 payments on a 10 year 3.8% loan, you'd pay it off in 10 years. if you did the same with a 20 yr 5.4% loan @ $600/month, you still have $56K left after 10 years. If you took the ~$300 / month savings and did nothing with it, just put it in the bank earning 0% you'd have $36,000 cash and $56,000 debt. So you're $20K behind after 10 years; that's the cost of extending IF you make the minimum payment only. If you used that $300 in after tax cash flow to put $300-$400 in your 457/403b/401k in vanguard wellesly (30% stocks 70% bonds, conservative given your cash poor indebted risk profile and available with your NJ 457 plan) and made 3% nominal, you'd have $40K-$53K ($300-$400) in a retirement account
So in ten years would you rather be debt free and have no assets. Or have $56K of 10 yr debt at 5.4% and a $40-$50K start to the investment nut. You'd still be behind the shorter term lower rate scenario, but not by that much $10K in 10 years is trivial in the context of your future earnings power and $300/month TODAY is a big deal.
In reality, I think you'll be much better off than either of those situations. Just saying it's not as simple as longer term higher rate = worse off